Which is better: praying Dhuhr or Asr with a friend on the examination committee, or delaying it to perform it in the mosque before its time expires?

If you hear the call to prayer and leaving would result in a significant, শরীয়ত-recognized harm, such as dismissal from a job that you need, then there is no impediment to praying inside the hall in congregation with your colleagues and students, while ensuring it is performed at the beginning of its time, due to the great reward associated with that. As stated in the Hadith: "Prayer at its appointed time."