Evaluating the Addition of Maintenance Immunotherapy Compared to the Usual Treatment of Chemotherapy and Autologous Stem Cell Transplant Alone for High-risk Patients With Relapsed or Refractory Hodgkin Lymphoma

Inclusion Criteria: * STEP 0 REGISTRATION: Patient must have biopsy confirmed relapsed classical Hodgkin lymphoma * STEP 0 REGISTRATION: Patient must be 5-75 years of age * STEP 0 REGISTRATION: Patient must have relapsed/refractory classical Hodgkin lymphoma (R/R cHL) after first line of chemotherapy * STEP 0 REGISTRATION: Patients \> 17 years of age must have an Eastern Cooperative Oncology Group (ECOG) performance status 0-2 and patients ≤ 17 years of age must have a Lansky performance status 50-100 * STEP 0 REGISTRATION: Patient must have had a PET CT or magnetic resonance imaging (MRI) (PET1) confirming relapse. The PET1 must have been completed prior to starting any salvage therapy and must be obtained within 56 days prior to Step 0 registration * NOTE: If patient received one cycle of salvage prior to study enrollment, PET1 confirming relapse must have been completed prior to the initiation of any salvage therapy * STEP 0 REGISTRATION: Patient must be considered standard- or high-risk at the time of initial relapse. If a patient meets one the following criteria below, they are considered high risk: * Primary refractory disease to frontline therapy * Relapse in \< 3 months after completion of frontline non-checkpoint inhibitor containing therapy * Relapse in \< 6 months after completion of frontline checkpoint inhibitor containing therapy * \> 4 disease sites at relapse (as defined by the German Hodgkin Study Groups \[GHSG\] Criteria) * Prior radiation that would result in overlapping fields that would exceed the RT dose to critical organs. For additional questions, contact the radiation oncology study co-chairs * Patients with bone marrow involvement * Patients who do not meet the criteria above are considered standard risk * STEP 0 REGISTRATION: Patient must be considered eligible for high dose chemotherapy and autologous hematopoietic cell transplant * STEP 0 REGISTRATION: 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 0 registration to rule out pregnancy * A patient of childbearing potential is defined as anyone, regardless of whether they have undergone tubal ligation, who meets the following criteria: * Has achieved menarche * Has not undergone a hysterectomy or bilateral oophorectomy * 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) * STEP 0 REGISTRATION: Patient 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 * STEP 0 REGISTRATION: 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 * STEP 0 REGISTRATION: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 (≤ 14 days prior to Step 0 registration) * STEP 0 REGISTRATION: Platelets ≥ 100,000/mm\^3 (≤ 14 days prior to Step 0 registration) * STEP 0 REGISTRATION: Total bilirubin ≤ 2 x institutional upper limit of normal (ULN) (≤ 14 days prior to Step 0 registration) * STEP 0 REGISTRATION: Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) ≤ 3.0 x institutional ULN (≤ 14 days prior to Step 0 registration) * STEP 0 REGISTRATION: Estimated glomerular filtration rate (GFR) (eGRF) ≥ 50 mL/min/1.73 m\^2 for patients \> 17 years of age (≤ 14 days prior to Step 0 registration) * STEP 0 REGISTRATION: Pediatric patients (\< 17 years old) must have one of the following: (≤ 14 days prior to Step 0 registration) * Estimated GFR (eGFR) ≥ 50 mL/min/1.73 m2 using the Bedside Schwartz formula (2009) * 24 hour urine creatinine clearance ≥ 50 mL/min/1.73 m\^2 * GFR ≥ 50 mL/min/1.73 m\^2 * Note: GFR must be performed using direct measurement with a nuclear blood sampling method OR direct small molecule clearance method (iothalamate or other molecule per institutional standard). * NOTE: Estimated GFR (eGFR) from cystatin C or other estimates not listed above are not acceptable for determining eligibility * STEP 0 REGISTRATION: Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months of Step 0 registration are eligible for this trial * STEP 0 REGISTRATION: For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated * STEP 0 REGISTRATION: 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 * STEP 0 REGISTRATION: Patient must not have any current or prior history of central nervous system (CNS) lymphoma * STEP 0 REGISTRATION: 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 * STEP 0 REGISTRATION: Patients must not have grade 2 or greater peripheral motor sensory neuropathy * STEP 0 REGISTRATION: Patient must not have a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or have current pneumonitis/interstitial lung disease * STEP 0 REGISTRATION: Patient must not have the following symptomatic autoimmune disorders: rheumatoid arthritis, systemic progressive sclerosis (scleroderma), systemic lupus erythematosus, Sjögren's syndrome, or autoimmune vasculitis (e.g., Wegener's granulomatosis), or conditions of immunosuppression that require current ongoing treatment with systemic corticosteroids (or other systemic immunosuppressants), including oral steroids (i.e., prednisone, dexamethasone). Patients who discontinue use of these classes of medication for at least 2 weeks prior to randomization eligible if, in the judgment of the treating physician investigator, the patient is not likely to require resumption of treatment with these classes of drugs during the study * Replacement doses of steroids for patients with adrenal insufficiency are allowed * Additionally, patients must not have an autoimmune disease that is felt by the treating physician to have the potential to be exacerbated by checkpoint inhibition * STEP 1 RANDOMIZATION: Patient must have undergone 2-4 cycles of salvage treatment as part of Step 0 of this protocol * STEP 1 RANDOMIZATION: Patients must be considered eligible for high dose chemotherapy and autologous hematopoietic cell transplant * STEP 1 RANDOMIZATION: Patient must have had a PET CT (PET2) following 2 cycles of on study salvage therapy and demonstrate complete metabolic response (CMR) or partial metabolic response (PMR). Patients with stable disease/no metabolic response (SD/NMR) or progressive metabolic disease (PMD) are ineligible to proceed to Step 1 randomization * STEP 1 RANDOMIZATION: Absolute neutrophil count (ANC) ≥ 1,000/mm\^3 (≤ 14 days prior to Step 1 randomization) * STEP 1 RANDOMIZATION: Platelets ≥ 75,000/mm\^3 (≤ 14 days prior to Step 1 randomization) * STEP 1 RANDOMIZATION: Total bilirubin ≤ 2 x institutional upper limit of control (ULN) (≤ 14 days prior to Step 1 randomization) * STEP 1 RANDOMIZATION: AST(SGOT) and ALT(SGPT) ≤ 3.0 x institutional ULN (≤ 14 days prior to Step 1 randomization) * STEP 1 RANDOMIZATION: Pediatric patients (\< 17 years old) must have one of the following: (≤ 14 days prior to Step 1 randomization) * Estimated GFR (eGFR) ≥ 50 mL/min/1.73 m\^2 using the Bedside Schwartz formula (2009) * 24 hour urine creatinine clearance ≥ 50 mL/min/1.73 m\^2 * GFR ≥ 50 mL/min/1.73 m\^2. GFR must be performed using direct measurement with a nuclear blood sampling method OR direct small molecule clearance method (iothalamate or other molecule per institutional standard) * NOTE: Estimated GFR (eGFR) from cystatin C or other estimates not listed above are not acceptable for determining eligibility * STEP 1 RANDOMIZATION: 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 randomization to rule out pregnancy * A patient of childbearing potential is defined as anyone, regardless of 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) * STEP 1 RANDOMIZATION: Patient 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 to use contraceptive measures for 5 months after the last dose of nivolumab and for 4 months after the last dose of pembrolizumab, as well as not breastfeed during these same timeframes