Does the governess have to repeat the Fajr prayer that she performed in congregation with a woman who started menstruating during the prayer, due to the invalidation of the Imam's prayer?

If it becomes clear to the Imam after the prayer that he was in a state of ritual impurity (hadath), the preponderant opinion is that the congregants (ma'mum) are not required to repeat their prayer, and only the Imam's prayer is invalidated. This is supported by the actions of a number of companions, such as Omar, Othman, Ali, and Ibn Omar, may Allah be pleased with them, who repeated their prayers but did not command the congregants to repeat theirs. What you did by continuing your prayer despite doubt about the discharge of menstrual blood is correct, because the default state is the continuation of purity. However, you must make up that prayer after you become pure from menstruation, according to many scholars.