All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
If you are well acquainted with the due times of prayers, do not pray before the due time. Hence, the calendar should not be relied upon if it contradicts the due time because offering the prayer after the beginning of its due time is a condition for its validity. Hence, no prayer is due unless such a condition is fulfilled. Those in charge of the calendar should correct this mistake through recognizing the regional differences of time. Marking the times of prayer is a trust for which they are responsible in front of Allaah The Exalted. To clear their consciences, at the very minimum, they should write proper statements so as to notify the people to recognize the differences in terms of prayer times. Otherwise, the Muslims have to recognize such differences out of their own responsibility.
Dear brother, adhere to gentleness in terms of advising people in your mosque so that they delay the prayer to its due time. Allaah The Exalted Says (what means): {Invite to the way of your Lord with wisdom and good instruction, and argue with them in a way that is best. Indeed, your Lord is most knowing of who has strayed from His way, and He is most knowing of who is [rightly] guided.} [Quran 16: 125].
However, if they do not respond to you, do not offer the prayer before the due time even if you would offer it alone in your house.
But if you are not well acquainted with the due times of prayers, so that your saying of the Athaan of Fajr being before the due time is based on assumption, something heard from others or just claims with no clear proofs, then you have to join the congregational prayer and avoid causing disorder for a matter you are not aware of. Allaah The Exalted Says (what means): {And do not pursue that of which you have no knowledge. Indeed, the hearing, the sight and the heart - about all those [one] will be questioned} [Quran 17: 36 ]
Allaah Knows best.