What is the ruling on some worshippers remaining in the mosque after Fajr prayer for individual dhikr, and others for reciting the Qur’an aloud in a group? And which is better: for dhikr to be in the mosque or at home?

It is recommended to remain in the mosque after the Fajr prayer until sunrise due to the great reward it entails, and because of the Prophet's (peace and blessings be upon him) saying: "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of a complete, complete, complete Hajj and Umrah (definition)."

As for reading the Quran collectively in one voice, it is not legislated. It becomes even more disliked if done in a loud voice that disturbs those praying and remembering Allah, based on his (peace and blessings be upon him) saying: "And let not some of you raise your voices over others in recitation during prayer." However, if the purpose of collective recitation is teaching, or if each person reads for themselves without causing disturbance, then it is permissible.