What is the correct time that should be followed for performing Fajr prayer in an area where the night is short in the summer, knowing that there is a difference between two times (1:33 and 2:43 after midnight), and is one sinful for missing congregational prayer due to the long distance and great hardship in going and coming back, knowing that he prays Friday prayer in the mosque?

You must rely on the calendar that aligns with the signs indicating the start of the legally prescribed prayer times, as each prayer has a specific time when it is not permissible to precede or delay it, except for a valid excuse. Fajr prayer begins with the appearance of true dawn (the widespread whiteness in the eastern horizon) and ends with sunrise. Whoever prays before the entry of its time, their prayer is not valid, and they must repeat it. If it becomes clear that a calendar they relied upon contradicts the legally prescribed prayer times, such that they prayed Fajr before its time, then they must repeat all the prayers they performed before their appointed time. It is advisable to inquire about prayer times from the knowledgeable people in the country. If the mosque is far and causes hardship, there is no sin in not going to it, and one can pray in congregation at home.