It is not permissible to lead people in Salah in a Masjid (mosque) that has a regular Imam except with his permission or in a case of necessity. If he is late, there is no harm to advance someone who is entitled to Imamah (leading congregational Salah) to lead people in Jumu`ah (Friday) Prayer. (Part No. 6; Page No. 279) When the Prophet (peace be upon him) was late,
Abu Bakr led the people in Salah. Also,
When the Prophet (peace be upon him) was late in Fajr (Dawn) Prayer in the Battle of Tabuk,
`Abdul-Rahman ibn `Awf (may Allah be pleased with him) led people in Salah. When the Prophet (peace be upon him) arrived, he found that
`Abdul-Rahman had led them in the first Rak`ah. He
(`Abdul-Rahman) wanted to step back but the Prophet (peace be upon him) pointed to him to complete his Salah. The Prophet (peace be upon him) offered the second Rak`ah behind him then offered the missed Rak`ah after Salam (end of prayer).
Muslim in his Sahih (book of sound Hadiths).May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.