Pembrolizumab and Brentuximab Vedotin vs GDP and Stem Cell Transplant for Relapsed/Refractory Hodgkin Lymphoma

Inclusion Criteria: * History of classic Hodgkin lymphoma by histopathology and now have relapsed or refractory disease after anthracycline-containing chemotherapy and eligible for high dose chemotherapy and autologous stem cell transplant * 18 years of age or greater * ECOG performance status 0-1 * Clinically and/or radiologically measurable disease as per the Lugano 2014 classification * Life expectancy \> 90 days * Absolute neutrophils ≥1.0 x 10\^9/L; Platelets ≥75 x 10\^9/L; Hemoglobin ≥80 g/L: Bilirubin ≤1.50 x UNL; AST and ALT ≤2.50 x UNL; Serum creatinine \<1.55 x UNL or Creatinine clearance ≥30 mL/min * Participant is able (i.e. sufficiently fluent) and willing to complete the quality of life questionnaires and/or health utility in either English or French * Participant consent must be appropriately obtained in accordance with applicable local and regulatory requirements. Each participant must sign a consent form prior to enrollment in the trial to document their willingness to participate. * Participants must be accessible for treatment and follow-up. * In accordance with CCTG policy, protocol treatment is to begin within 2 working days of participant enrollment * Women/men of childbearing potential must have agreed to use a highly effective contraceptive method during the study plus approximately 6 months after treatment completion * All patients must have a tumour block from their primary diagnostic biopsy and relapse/refractory biopsy if available and the centre/pathologist must have agreed to release the block or recently cut slides for correlative analysis if the participant has consented. If the primary diagnostic biopsy is not accessible, the original pathology report should be submitted for review and a biopsy from the relapse/refractory disease must be submitted. Exclusion Criteria: * Participants who have received prior salvage systemic therapy for their relapsed or refractory disease. * History of peripheral neuropathy or dyspnea ≥ grade 2 * Participants with a history of other malignancies except: adequately treated non-melanoma skin cancer and superficial bladder cancer, curatively treated in-situ cancer of the cervix or breast, or localized excised prostate cancer, other solid tumours curatively treated with no evidence of disease for \> 3 years * History of active CNS disease * Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy (at doses more than 10 mg prednisone or equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first and any dose of trial treatment * Has active autoimmune disease that has required systemic treatment in the past 3 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs) or history of allogeneic transplantation. Replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment * Known history of human immunodeficiency virus (HIV), active Hepatitis C Virus infection, active Hepatitis B Virus infection or any uncontrolled active systemic infection requiring intravenous (IV) antibiotics. Participants that are Hepatitis B core antibody positive are eligible if they are HBV DNA negative and are concurrently treated with anti-viral therapy. Participants with a past history of hepatitis C who have eradicated the virus are eligible * Clinically significant cardiovascular disease such as uncontrolled or symptomatic arrhythmias, congestive heart failure, angina, or myocardial infarction within 6 months of screening, or any Class 3 (moderate) or Class 4 (severe) cardiac disease as defined by the New York Heart Association Functional Classification * Documented history of cerebral vascular event (stroke or transient ischemic attack) * History of progressive multifocal leukoencephalopathy (PML). * Any serious active disease or co-morbid medical condition, including psychiatric illness, judged by the local investigator to preclude safe administration of the planned protocol treatment or required follow-up * Any other serious intercurrent illness, life-threatening condition, organ system dysfunction, or medical condition judged by the local investigator to compromise the subject's safety (for example): active, uncontrolled bacterial, fungal or viral infection; clinically significant cardiac dysfunction or cardiovascular disease * Participants who have been vaccinated with live, attenuated vaccines within 4 weeks of enrollment * Pregnant or lactating females, or women/men of childbearing potential not willing to use an adequate method of birth control for the duration of the study through 6 months after the last dose of trial treatment * Participants are not eligible if they have had a prior infusion reaction to the study drugs or their components \> grade 2 * Participant has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis * Participant has had an allogenic tissue/solid organ transplant * Concurrent or within the previous 4 weeks of randomization, treatment with other investigational drugs or anti-cancer therapy * Has received prior radiotherapy within 2 weeks of start of study treatment. Participants must have recovered from all radiation-related toxicities, not require corticosteroids, and not have had radiation pneumonitis. A one-week washout is permitted for palliative radiation (≤ 2 weeks of radiotherapy)