Is it permissible to pray Fajr in congregation in the mosque if the call to prayer (adhan) for it is given at the time of the false dawn (al-fajr al-kadhib)?

If you doubt the timing of Fajr adopted by the mosque-goers, and it is based on a calendar not approved by scholars, then do not pray the obligatory prayer with them. This is because prayer is not valid when there is doubt about the time having entered; rather, certainty or a strong presumption of its entry is required. Unknown calendars that have not been approved by trustworthy scholars can be used as an approximation to know the prayer time, but they cannot be relied upon entirely.