Answer:
The New Testament does not record that Peter took leadership of the early church. In fact, although Peter was undoubtedly an important leader, he most certainly did not have the pre-eminence. At the Jerusalem council in Acts 15, Peter was one of those who spoke, but it was James the brother of Jesus, who gave the final pronouncement and ruling with which all agreed. Thus historical records in the Bible itself, indicate that where it might have been expected that Peter would take a superior leadership role, he didn't, although he was most certainly mightily used of God in Acts 2 on the day of Pentecost.