Is it permissible for a latecomer to appoint himself as an imam for a group of latecomers after the first congregational prayer with their imam has ended?

If a latecomer (masbuq) stands up to complete the prayers he missed after his Imam finishes, it is permissible for others to follow him and take him as their Imam, especially if he caught at least one rak'ah with the original Imam. If he did not catch any rak'ah with the Imam, then it is permissible for him to be an Imam without any disagreement.