Must I complete the fast or break it if I took a cortisone medication after Fajr due to necessity to avoid kidney failure, knowing that I forgot to take it before Fajr?

First: If cortisone is taken orally after the dawn (Fajr) has genuinely broken, it invalidates the fast, and that day must be made up after Ramadan due to illness. However, if it is taken via injection (needle), it does not invalidate the fast unless it is for nourishment.

Second: Muezzins rely on astronomical calendars, which may contain errors in determining the time of Fajr prayer. The true dawn (Al-Fajr Al-Sadiq) may occur after the time specified in some calendars. Accordingly, if the medication was taken a short time after the Adhan, during which the dawn had not yet genuinely broken, the fast is valid. However, if a long time elapsed, the day must be made up. It is better to abstain from food and drink according to the calendar as a precaution.