Is it permissible to repeat the Fajr prayer at home as a precautionary measure if it was prayed in congregation at the mosque before the appearance of the white thread, for fear that the congregation might have prayed before the legitimate time began?

The time for Fajr prayer begins with the appearance of true dawn. The "white thread" and "black thread" mentioned in the verse refer to the whiteness of the day and the blackness of the night, and they can be distinguished by sight. Regarding an individual repeating a prayer after having performed it in congregation, out of fear that it might have occurred before its prescribed time, its ruling is mentioned in Fatwa (definition) no. 136776.