Is it permissible for the wife to pray the obligatory Fajr prayer in congregation behind her husband at home after he has prayed the Sunnah of Fajr, so that the husband may then go to pray the congregational Fajr prayer in the mosque?

The preponderant opinion is that it is permissible for someone performing a fard (obligatory) prayer to pray behind someone performing a nafl (supererogatory) prayer. This is based on the hadith of Mu'adh, who prayed the 'Isha prayer behind the Prophet (peace be upon him), then returned and led his people in prayer. This is the madhhab (school of thought) of Imam Al-Shafi'i. However, it is better not to do so because most scholars prohibit someone performing a fard prayer from praying behind someone performing a nafl prayer. Furthermore, congregational prayer is not obligatory for a wife in the first place, and the Salaf (early Muslims) used to perform the Raghibah (a specific supererogatory prayer) in their homes.