The fifth trump was called Le Pape in French and Il Papa in Italian, and he was exactly what the name says: the pope, paired with the female pope at the second trump, the two ecclesiastical powers matching the two secular ones. In a fifteenth-century Italian game that is unremarkable. In later contexts it became a problem.
The Besancon variant of the French pattern solved it by replacing both awkward cards with Roman deities: the Popess becomes Junon and the Pope becomes Jupiter. This was a Protestant and commercial adjustment for markets where a pope on a playing card was unwelcome, and it has nothing to do with esoteric doctrine, though it is occasionally cited as though it revealed a hidden pagan stratum. It reveals a printer avoiding trouble in a market where the Reformation had made a pope unsellable, which is an interesting fact about the card trade and not a fact about hidden doctrine.
It is worth being clear about what the original pairing meant. The bottom of the trump sequence in the fifteenth century is a picture of the social order: a mountebank, then a female pope, an empress, an emperor and a pope. Whatever else the trumps do, they begin by ranking the estates of the world, and the pope outranks the emperor. That is a statement, and it was a live one in fifteenth-century Italy.
Waite's solution was different and more consequential. He renamed the card the Hierophant, borrowing the title of the presiding priest of the Eleusinian mysteries, and thereby detached it from the Catholic Church without detaching it from institutional religion. The change stuck so thoroughly that most readers now encounter the card under a Greek title and never learn it was the pope. That is a loss, because the card's sharpest readings depend on his being a specific office with specific power over people, rather than a general symbol of tradition.