Inclusion Criteria:

          -  Incurable RM-HNSCC, defined as disease not amenable to cure by surgery and/or
             radiation therapy (or patient declines or is ineligible for surgery and/or radiation
             therapy).

          -  Disease Evaluation:

               -  Phase I: evaluable or measurable disease.

               -  Phase II: measurable disease, defined as lesions that can be accurately measured
                  in at least one dimension (longest diameter to be recorded) as ≥ 10 mm with CT
                  scan, as ≥ 20 mm by chest x-ray, or ≥ 10 mm by clinical exam.

          -  Prior Treatment:

               -  Phase I: any number of lines of prior therapy for RM-HNSCC.

               -  Phase I: prior therapy with inhibitors of CDK4/6 or PD-L1/PD-1 is acceptable.

               -  Phase II: RM-HNSCC that progressed or recurred within six months of
                  platinum-based therapy (given for curable or incurable disease).

               -  Phase II: prior therapy with inhibitors of CDK4/6 or PD-L1/PD-1 is not
                  acceptable.

          -  18 years of age or older

          -  Performance status 0-1 (ECOG)

          -  Adequate blood and organ function as defined:

               -  Absolute neutrophil count ≥ 1,500/mcL

               -  Platelets ≥ 100,000/mcL

               -  Hemoglobin ≥ 8.0 g/dL

               -  Total bilirubin ≤ 1.5 x ULN mg/dL

               -  AST(SGOT) ≤ 3 x IULN and ALT(SGPT) ≤ 3 x IULN

               -  Creatinine ≤ 2 x ULN OR creatinine clearance ≥ 40 mL/min/1.73 m2

               -  INR ≤ 1.5 x ULN and PTT ≤ 1.5 x ULN (Patients are allowed to be on
                  anticoagulation)

          -  Able to swallow oral medication

          -  Women of childbearing potential and men must agree to use adequate contraception
             (hormonal or barrier method of birth control, abstinence) beginning 14 days prior to
             first dose of abemaciclib, through the dosing period, and for at least 28 days after.

          -  Signed IRB approved written informed consent document.

        Exclusion Criteria:

          -  Phase II: prior inhibitors of CDK4/6 or PD-L1/PD-1 for treatment of incurable HNSCC.

          -  Radiation within 14 days of treatment start (patients who received radiotherapy must
             have completed and fully recovered from the acute side effects of radiotherapy),
             chemotherapy, targeted or investigational therapy within 21 days of treatment start.

          -  History of other malignancy ≤ 1 year prior to consent with the exception of completely
             resected skin carcinoma or other cancers with a low risk of recurrence.

          -  Ongoing toxicity attributed to prior anti-cancer therapy that is > grade 1, except
             alopecia or peripheral neuropathy

          -  Active central nervous system metastases: defined as currently receiving radiation
             therapy to metastatic CNS disease. Once radiation therapy is completed, patients with
             CNS disease are eligible if they meet all other criteria for enrollment.

          -  History of severe allergic reactions attributed to agents used in the study.

          -  Serious uncontrolled inter-current illness within the 3 months prior to study entry or
             psychiatric illness/social situations that would limit compliance with study
             requirements.

          -  Serious and/or uncontrolled preexisting medical condition(s) that, in the judgment of
             the investigator, would preclude participation in this study (i.e., interstitial lung
             disease, severe dyspnea at rest or requiring oxygen therapy, severe renal impairment
             [e.g. estimated creatinine clearance <30ml/min], 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).

          -  Active systemic bacterial infection (requiring intravenous [IV] antibiotics at time of
             initiating study treatment), fungal infection, or detectable viral infection (such as
             known human immunodeficiency virus positivity or with known active hepatitis B or C
             [for example, hepatitis B surface antigen positive]. Screening is not required for
             enrollment.

          -  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.

          -  Pregnant and/or breastfeeding. Patient must have a negative serum pregnancy test
             within 7 days of first dose of treatment.

          -  Active serious autoimmune disease requiring systemic immunosuppression (biologics,
             prednisone equivalent dose > 20 mg/day).

          -  Current use of strong CYP3A inhibitors or inducers.