Inclusion Criteria:
* Informed Consent: The patient must provide both oral and written informed consent.
* Diagnosis:
* Diagnosed with alcohol dependence according to the International Classification of Diseases, 10th Edition (ICD-10), and alcohol use disorder as per the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5).
* Diagnosed with schizophrenia spectrum disorder according to ICD-10 and DSM-5
* AUDIT Score: Alcohol Use Disorder Identification Test (AUDIT) score greater than 15.
* Body Mass Index (BMI): BMI of 23 kg/m² or higher.
* Age Range: Between 18 and 70 years old (inclusive).
* Heavy Alcohol Consumption: Defined as 4 or more heavy drinking days within a consecutive 21-day period during the 28 days preceding the baseline evaluation. The 21-day period will be selected based on the largest total alcohol consumption and the greatest number of heavy drinking days within the 28-day timeframe. This will be assessed using the Timeline Followback (TLFB) method. Heavy drinking days are defined as days with an alcohol intake of 4 or more units (48 g of alcohol) for women and 5 or more units (60 g of alcohol) for men.
Exclusion Criteria:
* Intellectual Disability: individuals with a diagnosis of intellectual disability.
* Acute Psychosis: Acute exacerbation of psychosis, as indicated by a score of 6 or 7 on the Clinical Global Impression-Severity (CGI-S) scale.
* Coercive Measures: Current use of coercive measures, which includes individuals sentenced to treatment ('dom til behandling').
* Suicidal Behaviour: Evidence of current severe suicidal behaviour, as assessed by the investigator during clinical evaluation.
* History of Severe Alcohol Withdrawal: History of delirium tremens or alcohol withdrawal seizures.
* Severe Withdrawal Symptoms: Clinical Institute Withdrawal Assessment of Alcohol Scale, revised (CIWA-Ar) score greater than 9 at baseline examination.
* Severe Neurological Conditions: Presence of severe neurological diseases, including severe traumatic brain injury.
* Diabetes: Type 1 or 2 diabetes
* Pregnant or Potentially Pregnant Women: WOCBP who are pregnant, breastfeeding, intend to become pregnant within the next 6 months (including 16 weeks of treatment plus two months after discontinuation of semaglutide), or are not using a highly effective contraceptive method throughout the study period. Highly effective methods include combined hormonal contraception (oral, intravaginal, transdermal), progestogen-only hormonal contraception (oral, injectable, implantable), intrauterine device (IUD), intrauterine system (IUS), bilateral tubal occlusion, vasectomised partner, or sexual abstinence. WOCBP with a measured serum human chorionic gonadotropin (hCG) level greater than 3 U/L at inclusion will also be excluded.
* Liver Function: Impaired hepatic function, defined as liver transaminases greater than three times the upper limit of normal.
* Renal Function: Impaired renal function, indicated by an estimated glomerular filtration rate (eGFR) below 50 mL/min and/or plasma creatinine above 150 μmol/L.
* Pancreatic Function: History of acute or chronic pancreatitis or amylase levels more than twice the upper limit of normal.
* Thyroid Conditions: Previous medullary thyroid carcinoma (MTC) or a family history of MTC and/or Multiple Endocrine Neoplasia syndrome type 2 (MEN 2).
* Cardiac Issues: Decompensated heart failure (NYHA class III or IV), unstable angina pectoris, or myocardial infarction within the past 12 months.
* Uncontrolled Hypertension: Systolic blood pressure above 180 mmHg or diastolic blood pressure above 110 mmHg.
* Alcohol Use Disorder Medication: Use of medications for alcohol use disorder (e.g., disulfiram, naltrexone, acamprosate, nalmefene) within the 28 days prior to inclusion as recorded in the Timeline Followback (TLFB) schedule.
* Investigational Drugs: Receipt of any investigational drug within the past three months.
* Weight-Lowering Medications: Use of other weight-lowering pharmacotherapy in the past three months.
* Allergic Reactions: Hypersensitivity to the active substance or any of the excipients.
* Language Barriers: Inability to speak and/or understand Danish.
* Other Conditions: Any other condition that, in the investigator\'s opinion, may interfere with participation in the trial.
For the subgroup of participants undergoing brain scans:
* MRI Contraindications: any contraindications for MRI (e.g., magnetic implants, pacemaker, claustrophobia).
* Benzodiazepine Use: Intermittent use of benzodiazepines within 12 days prior to the scanning session is not allowed. However, regular use of a stable dose of benzodiazepines is permitted.
