What is the ruling on someone who vomited before the Fajr adhan and drank intentionally after the adhan, having forgotten the prohibition of fasting? Is he required to make up the day (qada') or offer expiation (kaffara)?

If one is overcome by vomiting while fasting, their fast is not broken unless they deliberately re-swallow some of it into their stomach or deliberately induce the vomiting. In such cases, their fast is invalidated, and they must abstain from food and drink for the rest of the day and make up for that day later. They are not permitted to consume anything that breaks the fast. They must also repent, seek Allah's forgiveness, and make up the day without needing to offer expiation (kaffara).