Praise be to Allah.
If some of the members of the congregation were outside the mosque, and they were unable to follow the imam because the power cut out, then what is prescribed in this situation is for one of the worshippers who are near the door of the mosque to say the takbeer out loud so that those who are outside the mosque can hear and can follow the imam in the prayer. If he does not do that then they have the choice of either completing the prayer individually or sending one of their number forward to lead them in completing the prayer in congregation. This is better, so that the worshippers will not be confused and frustrated as a result of the imam’s voice being cut off.
Shaykh Ibn Baaz (may Allaah have mercy on him) was asked: A group of people prayed Jumu’ah in the basement of the mosque, and during the prayer the power cut out, and they could no longer hear the imam’s voice. One of them went forward and led them in the rest of the prayer. What is the ruling on their prayer, knowing that he completed the prayer as if it were Jumu’ah? What would be the ruling if no one went forward to lead them – should each person complete his prayer on his own? If that is permissible, should they complete it as Zuhr or as Jumu’ah, since they had listened to the khateeb and started to pray with the imam and prayed one rak’ah with him?
If the situation is as the questioner has described, then the prayer of all of them is valid, because whoever catches up with a rak’ah of Jumu’ah has caught up with Jumu’ah, as is stated in a saheeh hadeeth from the Messenger of Allaah (peace and blessings of Allaah be upon him). Even if no one went forward to lead them and each person prayed the second rak’ah by himself, that is sufficient, just as the one who joins the prayer late prays whatever he catches up with with the imam, then he makes up the other rak’ah by himself, because of the general meaning of the words of the Prophet (peace and blessings of Allaah be upon him): “Whoever catches up with a rak’ah of prayer has caught up with the prayer.” End quote from Majmoo’ Fataawa Ibn Baaz (12/331).
And Allaah knows best.