Christ did not give any temporal authority to Peter.
18 And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it. 19 And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose upon earth, it shall be loosed also in heaven. 20 Then he commanded his disciples, that they should tell no one that he was Jesus the Christ.
18 "Thou art Peter"... As St. Peter, by divine revelation, here made a solemn profession of his faith of the divinity of Christ; so in recompense of this faith and profession, our Lord here declares to him the dignity to which he is pleased to raise him: viz., that he to whom he had already given the name of Peter, signifying a rock, St. John 1. 42, should be a rock indeed, of invincible strength, for the support of the building of the church; in which building he should be, next to Christ himself, the chief foundation stone, in quality of chief pastor, ruler, and governor ; and should have accordingly all fulness of ecclesiastical power, signified by the keys of the kingdom of heaven.
Christ's three-fold mission - to teach, to heal and to rule