Is it permissible for some students to remain in the mosque after prayer to hold lessons while many lights are on, and is this considered extravagance, and what is the ruling on admonishing them about this?

Students should not be prevented from staying in the mosque to study. If the residents of the neighborhood pay the electricity bills, they should be gently reminded to contribute, and they will be rewarded. If the state covers the expenses, there is no justification for prevention. Lighting the lamps for studying is a simple matter and is not considered extravagance.