Inclusion Criteria:

          -  I1. Male or female patients aged ≥ 18 years at time of inform consent signature

          -  I2. Histologically proven metastatic or locally advanced HNSCC (oropharynx, oral
             cavity, hypopharynx and larynx). Patients with cancer of nasopharynx (i.e. cavum
             cancer) are not eligible

          -  I3. Availability of a representative formalin-fixed paraffin-embedded (FFPE) primary
             and/or metastatic tumor tissue with an associated pathology report for molecular
             pre-screening: either an archival tumor block or a dedicated freshly collected tumor
             biopsy.

          -  I4. Documented CDKN2A homozygous deletion and/or CCND1 amplification and/or CDK6
             and/or CDK4 amplification and no deletion/losses more than single copy of RB1 by copy
             number data before C1D1.

        Note: This molecular pre-screening will be centralized at at the CGH platform of Centre
        Léon Bérard (CLB).

        Note: This molecular pre-screening will be centralized at the CGH platform of Centre Léon
        Bérard (CLB).

        Note: This molecular pre-screening can be performed for patient without documented disease
        progression (PD) but study drug treatment cannot be initiated until confirmed radiological
        PD.

          -  I5. HPV negative tumor status must be documented before C1D1. Note: This analysis will
             be centralized and performed by translational Biopathology platform of CLB during
             molecular pre-screening by IHC for p16.

          -  I6. Documented radiological progression or relapse after at least platin and cetuximab
             or anti-EGFR-based chemotherapy (combination or sequential treatment) and other
             standard treatment available at time of C1D1..

          -  I7. At least one measurable lesion by CT-scan as per RECIST 1.1.

          -  I8. At least one biopsiable tumor lesion before C1D1 i.e. at least one lesion with a
             diameter of at least 10 mm, visible by medical imaging and accessible to percutaneous
             sampling.

          -  I9. Eastern Cooperative Oncology Group (ECOG) performance status: 0 or 1.

          -  I10. Life expectancy > 12 weeks.

          -  I11. Patients must be able to swallow capsules.

          -  I12. Adequate organ and bone marrow function as defined by the following tests (to be
             checked using medical records and then carried out within 7 days prior C1D1):

               -  Bone marrow :

                    -  Absolute neutrophil count >= 1.0 x 109/L

                    -  Platelet count > 100 x 109/L

                    -  Hemoglobin value >= 9 g/dL Note : 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.

               -  Renal function

                    -  Calculated creatinine clearance by MDRD or CDK-EPI > 50mL/min/1.73m2

               -  Liver function

                    -  Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 3 x
                       ULN (or < 5.0 x ULN if liver metastases are present)

                    -  Total serum bilirubin ≤ 1.5 x ULN (except for patients with Gilbert disease
                       for whom a total serum bilirubin ≤ 3mg/dL is acceptable).

          -  I13. Women of childbearing potential (entering the study after a confirmed menstrual
             period and who have a negative pregnancy test within 7 days before C1D1) must agree to
             use two methods of medically acceptable forms of contraception from the date of
             negative pregnancy test up to 3 months after the last study drug intake.

          -  I14. Fertile males must use a highly effective contraception during dosing period and
             through 3 months after final dose of study drug.

          -  I15. Patient should be able and willing to comply with study visits and procedures as
             per protocol.

          -  I16. Patient should understand, sign, and date the written voluntary informed consent
             form at the screening visit prior to any protocol-specific procedures performed.

          -  I17. Patients must be covered by a medical insurance.

        Exclusion Criteria:

          -  NI1. Cancer disease considered curable with surgery or radiotherapy.

          -  NI2. Patient with a concurrent malignancy or has a malignancy within 3 years of study
             enrollment (with the exception of adequately treated basal or squamous cell carcinoma
             or non-melanomatous skin cancer).

          -  NI3. Patient with impairment of gastrointestinal (GI) function or GI disease that may
             significantly alter the absorption of abemaciclib (e.g., ulcerative diseases,
             uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome, or small bowel
             resection).

          -  NI4. Patient has other concurrent severe and/or uncontrolled medical condition that
             would, in the investigator's judgment contraindicate her participation in the clinical
             study (for example, history of major surgical resection involving the stomach or small
             bowel, or preexisting Crohn's disease or ulcerative colitis).

          -  NI5. Persisting significant toxicities related to prior treatments i.e. ≥ Grade 2 AE
             according to CTCAE V5.0 except alopecia (any grade), grade 2 peripheral neuropathy and
             biological values as defined in inclusion criteria.

          -  NI6. Hypersensitivity to the active substance or excipient of study drug.

          -  NI7. Have received prior treatment with any CDK4/6 inhibitor (or participated in any
             CDK4/6 inhibitor clinical trial for which treatment assignment is still blinded).

          -  NI8. Patient has received treatment with a drug that has not received regulatory
             approval for any indication within :

               -  14 days of C1D1 for non myelosuppressive agent or

               -  21 days of C1D1 for a myelosuppressive agent.

          -  NI9. Patient has had major surgery and/or radiotherapy within 14 days prior to C1D1.

          -  NI10. Patient has received within 28 days prior to C1D1 yellow fever vaccine.

          -  NI11. Patient has a personal history within the last 12 months prior to C1D1 of any of
             the following conditions: syncope of cardiovascular etiology, ventricular tachycardia,
             ventricular fibrillation, or sudden cardiac arrest.

          -  NI12. Patient needs for the following concomitant medications/interventions not
             permitted during the study treatment period:

               -  Any investigational anticancer therapy other than the study drug.

               -  Any concurrent chemotherapy, radiotherapy (except palliative radiotherapy after
                  discussion with the Sponsor), immunotherapy, biologic or hormonal therapy for
                  cancer treatment. Concurrent use of hormones for non-cancer-related conditions
                  (e.g., insulin for diabetes and hormone replacement therapy) is acceptable.

               -  Major surgery.

               -  Strong and moderate inducers and inhibitors of CYP3A (for example grapefruit or
                  grapefruit juice, phenytoin and carbamazepine).

               -  Enzyme-Inducing Anti-Epileptic Drugs (EIAED).

          -  NI13. Patient has received an autologous or allogeneic stem-cell transplant.

          -  NI14. Patient has an active systemic fungal and/or known viral infection (for example,
             human immunodeficiency virus antibodies, hepatitis B surface antigen, or hepatitis C
             antibodies).

          -  NI15. Pregnant or breast-feeding female patients.

          -  NI16. Has known active central nervous system (CNS) metastases and/or carcinomatous
             meningitis. Patients with previously treated brain metastases may participate provided
             they are stable (without evidence of progression by imaging for at least four weeks
             prior to C1D1and any neurologic symptoms have returned to baseline), have no evidence
             of new or enlarging brain metastases, and are not using steroids for at least 28 days
             prior to C1D1. This exception does not include carcinomatous meningitis which is
             excluded regardless of clinical stability