Is it permissible to forgo congregational prayer (Salat al-Jama'ah) for Fajr due to short sleep time and the need to focus on studies and exams?

Congregational prayer in the mosque has great virtue and reward, but it is not obligatory. It is permissible anywhere, and the reward is attained thereby. So, if you wake up before the time expires and pray in congregation with your family, there is no sin upon you.