He commissioned, Peter (feed my lambs, feed my sheep) and his successors for this task.
Jesus commissioned ALL of the Apostles to spread the Word. And there is no word that Peters "successors" (where is that verse found using that wording?) are any different from the other Apostles followers.
He founded one Church (NOT Catholic!), empowered it (Wasn't He was speaking directly to the Disciples) with a binding authority and assured that the gates of hell shall never prevail against it (This is the Heavenly Church consisting of saved saints. Any man made denomination IS subject to failure).
The Churchs the Disciples started were based on Jewish religion -starting with the Ten Commandments. The one that says keep the Sabbath day holy. It may also forbid graven images - quick hide the statue of Mary! That's just 2 of the differences between the RCC and the Churchs that Peter and the others started.
No mention of Peter specifically nor as head over the other apostles.