STEP 1 REGISTRATION:

Patient must be >= 18 years of age

Patient must have biopsy-proven metastatic squamous cell carcinoma, originating in the oral cavity, larynx, oropharynx, or hypopharynx, with active disease present in both the head and neck and distant sites
* NOTE: The tumor from an oropharynx primary site must have known p16 status; p16 positive cancer of unknown primary is allowed as well, provided the disease presentation is consistent with a head and neck primary

Patient can have prior surgical resection of a primary cancer in the head and neck at any previous time, however, residual/recurrent disease in the head and neck must be present on baseline imaging

Patients must not have prior head and neck radiotherapy

Any effects from prior cancer therapy for other diseases must be fully resolved and not pose a problem for giving the treatment on this trial

Patient must have 4 or fewer metastatic sites prior to starting any treatment, with thoracic nodal disease considered a single site if encompassable in a tolerable radiotherapy hypofractionated field (i.e.,15 fractions or less)
* NOTE: Contiguous/adjacent metastases treatable in a single stereotactic field may be considered a single site
* NOTE: Patients with additional indeterminate findings such that the total number of metastatic sites would be more than 4 may be enrolled if a non-malignant etiology to these findings is a reasonable consideration

Patient must have Eastern Cooperative Oncology Group (ECOG) performance status 0-1

Patient must have the ability to understand and the willingness to sign a written informed consent document. Patients with impaired decision-making capacity (IDMC) who have a legally authorized representative (LAR) or caregiver and/or family member available will also be considered eligible

Patients must have measurable disease as follows:
* For patients who have not started any initial systemic therapy (with pembrolizumab + chemotherapy) must have measurable disease documented by CT of the neck and chest, and abdomen obtained within 28 days prior to Step 1 registration
* For patients who have started or completed their 3 cycles of initial systemic therapy (with pembrolizumab + chemotherapy) must have measurable disease documented by CT of the neck, chest and abdomen obtained within 28 days prior to the start of their initial systemic therapy

Leukocytes >= 3,000/mcL (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Absolute neutrophil count (ANC) >= 1,500/mcL (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Platelets >= 100,000/mcL (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Total bilirubin =< institutional upper limit of normal (ULN). Patients with a total bilirubin > 1.5 x ULN, that is attributed to confirmed Gilbert's syndrome, are allowed after consultation and approval from their treating physician (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT)(serum glutamic pyruvic transaminase [SGPT]) =< 3.0 x institutional ULN (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Creatinine clearance: Glomerular filtration rate (GFR) >= 50 mL/min/1.73m^2 (for patients receiving carboplatin-based regimens, GFR > 30 mL/min/1.73m^2) (obtained =< 28 days prior to Step 1 registration or prior to the start of any chemotherapy if on Arm T)

Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of Step 1 registration are eligible for this trial

For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated

Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load

Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial

Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better

Patient must not have an active autoimmune disease (i.e., inflammatory bowel disease, systemic lupus erythematosus, rheumatoid arthritis, etc.) that has required systemic treatment (i.e., disease modifying agents, corticosteroids, or immunosuppressive drugs) in past 2 years. Replacement therapy (i.e., thyroxine, insulin, physiologic corticosteroid replacement) is not considered a form of systemic treatment and is allowed

Patients on Arm S must have received chemoimmunotherapy

Patients will be enrolled in the quality of life (QOL) study if the patient can read and understand English, Spanish, French or Chinese (simplified or traditional characters)
* NOTE: Sites cannot translate the associated QOL forms

Patient must not be pregnant or breast-feeding due to the potential harm to an unborn fetus and possible risk for adverse events in nursing infants with the treatment regimens being used. All patients of childbearing potential must have a blood test or urine study within 14 days prior to Step 1 registration to rule out pregnancy. A patient of childbearing potential is defined as anyone, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy; or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months)

Patients must not expect to conceive or father children by using accepted and effective method(s) of contraception or by abstaining from sexual intercourse for the duration of their participation in the study. Patients of childbearing potential must continue contraceptive measures for 4 months after the last dose of protocol treatment and must not breastfeed while on study treatment through 4 months after the last dose of protocol treatment

Patient must not have received any live vaccine within 30 days prior to Step 1 registration and while participating in the study. Live vaccines include, but are not limited to, the following: measles, mumps, rubella, chicken pox, yellow fever, rabies, bacillus Calmette Guerin (BCG), and typhoid (oral) vaccine. Patients are permitted to receive inactivated vaccines and any non-live vaccines including those for the seasonal influenza and coronavirus disease 2019 (COVID-19) (Note: intranasal influenza vaccines, such as Flu-Mist [trademark] are live attenuated vaccines and are not allowed). If possible, it is recommended to separate study drug administration from vaccine administration by about a week (primarily, in order to minimize an overlap of adverse events)

STEP 2 RANDOMIZATION:

Patient must have ECOG performance status 0-2

Patient must have completed 3 cycles of initial systemic chemotherapy

For patients registered to Arm S on Step 1, patients must have at least stable disease after completing 3 cycles of pembrolizumab + chemotherapy

Patient must have no signs of progression (complete response [CR]/partial response [PR] or stable disease [SD]) on restaging imaging (consisting of neck, chest, and abdomen CT). Restaging imaging must have been done after completion of initial systemic chemotherapy with pembrolizumab + chemotherapy on Step 1 and within 7 days prior to Step 2 randomization. Patients with stable or responding radiologic response are eligible for Step 2

PRIMARY OBJECTIVE: 
I. To compare overall survival (OS) between immunotherapy plus consolidative radiotherapy (CoRT) and immunotherapy alone following non-progression with systemic chemoimmunotherapy.

SECONDARY OBJECTIVES:
I. To compare progression-free survival (PFS) between the two arms.
II. To compare time-to-treatment failure (TTF) between the two arms. 
III. To determine the risk of non-hematologic high-grade (3 or higher) toxicity with the addition of CoRT. 
IV. To establish the prognostic value of quantitative positron emission tomography (PET) biomarkers at baseline (standardized uptake value maximum [SUVmax], metabolic tumor volume [MTV], total lesion glycolysis [TLG]) for overall survival in both arms.
V. To establish the predictive value of (a) structured qualitative read (Hopkins Criteria) and (b) quantitative analysis for assessment of the post-radiotherapy or chemotherapy restaging PET/computed tomography (CT) to evaluate its association with overall survival in both arms.

HEALTH-RELATED QUALITY-OF-LIFE (HRQL) OBJECTIVES:
I. To compare the time-to-definitive-deterioration (TTDD) between the two arms. (PRIMARY)
II. To compare the mean early change in the Functional Assessment of Cancer Therapy – Head & Neck (FACT-HN) trial outcome index (TOI) between the arms, defined as the difference between the cycle 7 time point and randomization. (SECONDARY)
III. To compare the time-to-deterioration (TTD) between the arms (first deterioration). (SECONDARY)
IV. To compare the nadir of the Functional Assessment of Cancer Therapy-Immune Checkpoint Modulator (FACT-ICM) score over the course of study participation between the arms. (EXPLORATORY)
V. To compare quality-adjusted survival between the arms. (EXPLORATORY)

EXPLORATORY OBJECTIVES:
I. To identify differences in patterns-of-failure with respect to local regional and distant recurrences following CoRT versus immunotherapy alone. 
II. To evaluate the risk of tracheostomy and/or gastrostomy in patients treated with CoRT versus immunotherapy alone.

OUTLINE:

STEP 1: Patients who have not completed initial systemic therapy prior to enrollment are assigned to Arm T and patients who have completed initial systemic therapy prior to enrollment are assigned to Arm S.

ARM T: Patients receive 1 of 3 regimen options: Option 1: Pembrolizumab intravenously (IV) on day 1 of each cycle or every other cycle with carboplatin IV on day 1 of each cycle or days 1 and 8 of each cycle and paclitaxel on day 1 of each cycle or days 1 and 8 of each cycle; Option 2: Pembrolizumab IV on day 1 of each cycle or every other cycle with cisplatin IV on day 1 of each cycle and fluorouracil IV on days 1-4 of each cycle; Option 3: Pembrolizumab IV on day 1 of each cycle or every other cycle with carboplatin IV on day 1 of each cycle and fluorouracil IV on days 1-4 of each cycle. Treatment repeats every 21 days for up to 3 cycles in the absence of disease progression or unacceptable toxicity.

ARM S: Patients proceed directly to Step II.

STEP II: Patients are randomized to 1 of 2 arms.

ARM A: Patients receive one cycle of initial regimen from Arm T. Patients then receive one of the following: 1) Pembrolizumab IV on day 1 of each cycle. Cycles repeat every 21 days for 2 years of total treatment in the absence of disease progression or unacceptable toxicity. 2) Pembrolizumab IV on day 1 of each cycle. Treatment repeats every 21 days for 2 cycles in the absence of disease progression or unacceptable toxicity. Patients then receive pembrolizumab IV on day 1 of each cycle. Cycles repeat every 6 weeks for 2 years of total treatment in the absence of disease progression or unacceptable toxicity. Patients also receive radiation therapy once daily (QD) for a total of 30 fractions. In addition, patients undergo CT, PET/CT, and/or magnetic resonance imaging (MRI) throughout the trial. 

ARM B: Patients receive one cycle of initial regimen from Arm T. Patients then receive one of the following: 1) Pembrolizumab IV on day 1 of each cycle. Cycles repeat every 21 days for 2 years of total treatment in the absence of disease progression or unacceptable toxicity. 2) Pembrolizumab IV on day 1 of each cycle. Treatment repeats every 21 days for 2 cycles in the absence of disease progression or unacceptable toxicity. Patients then receive pembrolizumab IV on day 1 of each cycle. Cycles repeat every 6 weeks for 2 years of total treatment in the absence of disease progression or unacceptable toxicity. Patients also undergo CT, PET/CT, and/or MRI throughout the trial.

After completion of study treatment, patients are followed up for 3 years.