It's tricky. The Catholic Church claims to be the original Christian religion, with the idea that the Popes form a chain of leaders that go all the way back to Peter the apostle as the first leader of the Church.

But it all depends on your perspective. In reality (or at least my opinion), it's more of a family tree of sects branching off from each other that all go back to the original followers of Christ. Of course, ancient Christianity would have looked very different from all modern religions, so it's a bold claim for any to say it is the same as that original group.