Abemaciclib and Pembrolizumab in Metastatic or Recurrent Head and Neck Cancer

Inclusion Criteria: * Histologically confirmed (core biopsy proven) metastatic or recurrent squamous cell carcinoma of head and neck * Adequate pulmonary and cardiac function * Available archived tissue of primary tumor or resected tumor specimen with adequate samples * Prior treatment with immune checkpoint inhibitor is not allowed in cohort 1 patients. Patients in cohort 2 should have failed or progressed on prior immune checkpoint inhibitor * Eastern Cooperative Oncology Group Performance Status(ECOG PS) = 0 or 1 * Patients who received chemotherapy must have recovered (Common Terminology Criteria for Adverse \[CTCAE\] Grade \<= 1) from the acute effects of chemotherapy except for residual alopecia or Grade 2 peripheral neuropathy prior to randomization. A washout period of at lease 21 days is required between last chemotherapy dose and randomization (provided the patient did not receive radiotherapy) * Patients who received adjuvant radiotherapy must have completed and fully recovered from the acute effects of radiotherapy. A washout period of at least 14 days is required between end of radiotherapy and randomization * The patient is able to swallow oral medications * Adequate hematologic and end-organ function * Absolute Neutrophil Count (ANC) \>= 1500/mm3 * Platelet count ≥ 100,000/mm3 * Hemoglobin (Hb) ≥ 8g/dl (Patients may receive erythrocyte transfusions to achieve this hemoglobin level at the discretion of the investigator. Initial treatment must not begin earlier than the day after the erythrocyte transfusion) * Creatinine (Cr) ≤ 1.5 x Upper Limit of Normal (ULN) or Creatinine Clearance (CrCl) ≥ 60 ml/min * Total Bilirubin ≤ 1.5 x ULN (except subjects with Gilbert syndrome, who can have total Bilirubin \< 2.0 x ULN and direct bilirubin within normal limits are permitted.) * Aspartate Aminotransferase (AST) and Alanine aminotransferase (ALT) and alkaline phosphatase ≤ ULN * Agreement to remain abstinent or use appropriate contraception, among women of childbearing potential * Willingness and ability to consent for self to participate in study * Willingness and ability to comply with scheduled visits, treatment plan, laboratory tests, and other study procedures Exclusion Criteria: * Autoimmune disease (Note: Vitiligo, type 1 diabetes mellitus, residual hypothyroidism due to autoimmune thyroiditis only requiring hormone replacement, and conditions not expected to recur in the absence of an external trigger are permitted.) * Condition requiring systemic treatment with either corticosteroids (\>10 mg daily prednisone equivalent) or other immunosuppressive medications within 14 days prior to study treatment (Note: Inhaled and topical steroids, and adrenal replacement steroid doses \> 10 mg daily prednisone equivalent, are permitted in the absence of active autoimmune disease.) * Preexisting medical condition(s) that would preclude participation in this study (for example, interstitial lung disease, severe dyspnea at rest or requiring oxygen therapy, history of major surgical resection involving the stomach or small bowel, or preexisting Crohn's disease or ulcerative colitis or a preexisting chronic condition resulting in baseline Grade 2 or higher diarrhea). * Immunosuppression, of any kind * Prior treatment with Cyclin-Dependent Kinase(CDK) 4/6 Inhibitor * Major surgical procedure or significant traumatic injury within 4 weeks prior to study treatment, and must have fully recovered from any such procedure * Personal history of any of the following conditions: syncope of cardiovascular etiology, ventricular arrhythmia of pathological origin (including, but not limited to, ventricular tachycardia and ventricular fibrillation), or sudden cardiac arrest * Angina, myocardial infarction (MI), symptomatic congestive heart failure, cerebrovascular accident, transient ischemic attack TIA), arterial embolism, pulmonary embolism, percutaneous transluminal coronary angioplasty (PTCA), or coronary artery bypass grafting (CABG) within 6 months prior to study treatment * Known active viral or non-viral hepatitis or cirrhosis * Any active infection requiring systemic treatment, positive tests for Hepatitis B surface antigen or Hepatitis C ribonucleic acid (RNA). * History of gastrointestinal perforation or fistula in the 6 months prior to study treatment, unless underlying risk has been resolved (e.g., through surgical resection or repair) * Known human immunodeficiency virus (HIV) or acquired immunodeficiency syndrome (AIDS) related illness * Pregnancy or breastfeeding - Female patients must be surgically sterile (i.e., ≥6 weeks following surgical bilateral oophorectomy with or without hysterectomy or tubal ligation) or be postmenopausal, or must agree to use effective contraception during the study and for 4 months following last dose of treatment. All female patients of reproductive potential must have a negative pregnancy test (serum or urine) within 7 days prior to study treatment. Male patients must be surgically sterile or must agree to use effective contraception during the study and for 4 months following last dose of treatment. * Other severe acute or chronic medical or psychiatric condition or laboratory abnormality that may increase the risk associated with study participation or may interfere with the interpretation of study results and, in the judgment of the Investigator, would make the patient inappropriate for this study