Is it obligatory to attend the congregational Fajr prayer in the mosque if that leads to performing it at night, or is it permissible to pray it at home after the time for it has begun?

The time for Fajr begins with the true dawn (the horizontal white light across the horizon) and extends until sunrise. Most calendars do not accurately determine the correct time for the true dawn. Therefore, the people of each town should investigate the time of Fajr and inform others. If you doubt performing the prayer before its time, wait until you are reasonably certain or sure that the time has begun.

The mosque congregants should be advised to delay the prayer until they are reasonably certain the time has begun. If they do not respond, look for another mosque. If you cannot find one, pray with them in the mosque so as not to be thought ill of and not to deprive yourself of the reward of walking to mosques. Then, repeat the prayer at home with your family after the time has begun.