Women who pray at home can pray before the Athan if they are certain the time is due. This masjid most likely is following the hanafi school of thought and delaying prayer like this is not from the Sunnah.

Mn should try to pray in another masjid that follows the Sunnah. If this is not possible, they should pray with them.

Asr time starts when the shadow of an object is equal it’s size. If this is right then the actual time of Asr starts at 4.06 pm and Fajr at 5.20 am but in our locality Asr is delayed to 5.30pm and Fajr to 6.00am. In this case what are the ruling for the men who join the jamath and the women who pray at home?