Praise be to Allah.
If the situation is as you describe, then when they enter the mosque it is prescribed for them to pray Tahiyyat al-Masjid before sitting down. The imaam may order that the Iqaamah be given for Fajr prayer, after which there is no need for Tahiyyat al-Masjid, because praying the fard (obligatory) prayer is sufficient and means that there is no longer any need to do Tahiyyat al-Masjid. The member of the congregation may also forego Tahiyyat al-Masjid if he enters the mosque when the Iqaamah is being given.
And Allaah is the source of strength. May Allaah bless our Prophet Muhammad and his family and companions.