Does a woman receive the reward for congregational prayer if she prays Fajr in her home, and is the husband sinful if he leads her in Fajr prayer after he has prayed it himself?

A woman's prayer in her home is better than her prayer in the mosque, and the best place for her prayer is her private chamber (a small room within a larger room). Similarly, a man's prayer does not suffice for his wife, because purely physical acts of worship, such as prayer, do not permit representation by proxy, according to the unanimous agreement of scholars.