After Peter died Linus claimed to be the head of the Church
Actually, there is no record of Linus claiming to be anything. What is recorded is that in 180, Irenaeus wrote that the apostles committed to Linus the office of the episcopate.
"Actually, there is no record of Linus claiming to be anything. What is recorded is that in 180, Irenaeus wrote that the apostles committed to Linus the office of the episcopate."
Linus was made a Bishop by the Apostles, which was the leader of a local congregation.
Somewhere down the road the Catholic church decided the head of the church would be the "Bishop of Rome".
List of Popes
Peter was an Apostle. Linus was not. If you look at the list provided by New Advent, John was probably still on the earth even through Pope Clement I.
Linus, Anacletus, and Clement (I) literally could
NOT be head of the true Church of Christ, one because they never had the authority and two, a higher authority was
proven to be on the earth at the same time.