A Study to Test Whether Different Combinations of BI 765063, Ezabenlimab, Chemotherapy, Cetuximab, and BI 836880 Help People With Head and Neck Cancer or Liver Cancer

Inclusion Criteria:

* Signed informed consent form (ICF) prior to any trial-specific procedures.
* Male or female aged ≥ 18 years at the time of ICF signature.
* Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 at the screening visit.
* Expected life expectancy of at least 3 months.
* Patients homozygous for V1 allele (including V1-like alleles) of Singal Regulatory Protein-alpha (SIRPα) (V1/V1 SIRPα genotype). SIRPα polymorphism will be assessed in blood sampling (using patient Deoxyribonucleic Acid \[DNA\]) during Screening 1 Visit.
* Patients with at least one measurable lesion as per Response Evaluation Critiera In Solid Tumours (RECIST) version 1.1 (v1.1).
* Patients must agree to provide a mandatory pre-treatment (baseline) biopsy and an ontreatment fresh tumour biopsy (unless medically contraindicated. or mandatory requirement is lifted by the sponsor). Details on biopsy sample collection are provided in the Lab Manual.

  \-- Pre-treatment (baseline) biopsy: A fresh tumour biopsy before receiving the trial medication is preferred. In case a fresh tumour biopsy cannot be obtained, the Sponsor must be notified and archival formalin-fixed paraffin embedded (FFPE) tumour tissue block from the most recent time point before entering the trial must be provided (maximum 6 months prior to study entry). If these requirements cannot be met, the patient may still be allowed to enter the study at the discretion of the sponsor, after discussion between the Investigator and Sponsor.
* Female patients. Women of childbearing potential (WOCBP) must agree to use highly effective methods of contraception per ICH M3 (R2), that results in a less than 1% per year failure rate when used consistently and correctly, starting at the screening visit, during the trial and for 6 months after the end of trial treatment. The requirement of contraception does not apply to women of no childbearing potential, but they must have evidence of such at screening. Women of childbearing potential must have a serum negative pregnancy test within 72 hours prior to first drug administration.

Women who are postmenopausal for at least 1 year (defined as more than 12 months since last menses) or are surgically sterilized do not require this test. The following methods of contraception are considered highly effective:

* Combined (oestrogen and progestogen containing) hormonal birth control that prevents ovulation (oral, intravaginal, transdermal)
* Progestogen-only hormonal birth control that prevents ovulation (oral, injectable, implantable)
* Intrauterine device (IUD) or intrauterine hormone-releasing system (IUS)
* Bilateral tubal occlusion
* Vasectomised partner (provided that this is the sole sexual partner and has received medical assessment of the surgical success)
* Sexual abstinence (if accepted by local ethics boards and regulatory agencies as highly effective) Sexual abstinence is defined as refraining from heterosexual intercourse during the entire study period from screening through 6 months after last trial treatment. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the subject.

Periodic abstinence (calendar, symptothermal, post-ovulation methods) is not an acceptable method of contraception for use during this trial.

Although use of a contraceptive pill and Intrauterine device (IUD) together are considered a highly- effective method of birth control, women of childbearing potential taking a contraceptive pill must use an additional barrier method for the entire duration of the trial treatment intake and for 6 months after the end of the trial treatment intake. WOCBP is defined as: fertile, following menarche and until becoming post-menopausal unless permanently sterile. Permanent sterilisation methods include hysterectomy, bilateral salpingectomy and bilateral oophorectomy. Tubal ligation is NOT a method of permanent sterilisation. A postmenopausal state is defined as no menses for 12 months without an alternative medical cause.

Further inclusion criteria apply.

Exclusion criteria:

* Patients with at least one SIRPα V2 allele, i.e. SIRPα V1/V2 or V2/V2 individuals.
* Patients with symptomatic/active central nervous system (CNS) metastases. Patients with previously treated brain metastases are eligible, if there is no evidence of progression for at least 28 days before the first trial drug administration without requirement for treatment with corticosteroids, as ascertained by clinical examination and brain imaging (MRI (magnetic resonance imaging) or CT (computed tomography)) during the screening period.
* Prior allogeneic stem cell or solid organ transplantation.
* Any tumour location necessitating an urgent therapeutic intervention (e.g., palliative care, surgery or radiation therapy, such as spinal cord compression, other compressive mass, uncontrolled painful lesion, bone fracture).
* Presence of active invasive cancers other than the one treated in this trial within 5 years prior to screening, except appropriately treated basal cell carcinoma of the skin, or in situ carcinoma of uterine cervix, or other local tumours considered cured by local treatment.
* Patients with active autoimmune disease or a documented history of autoimmune disease, that requires systemic treatment, i.e. corticosteroids or immunosuppressive drugs, except patients with vitiligo, resolved childhood asthma/atopy, alopecia, or any chronic skin condition that does not require systemic therapy; patients with autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone and/or controlled Type 1 diabetes mellitus on a stable insulin regimen are eligible.
* History of severe hemorrhagic or thromboembolic event in the past 12 months (excluding central venous catheter thrombosis, peripheral deep vein thrombosis and portal vein thrombosis due to HCC tumor invasion).
* Known prior history of severe infusion related reactions to monoclonal antibodies (Grade ≥ 3 National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events (CTCAE) v5.0).

Further exclusion criteria apply.