The P most likely stands for Pope/Papacy, but would reflect the Vatican and the hierarchy of the Catholic Church, not the average Catholic practitioner. The Catholic Church was established like all of the major religions by The Families as a means of control over the masses. ((They)) have controlled the Catholic Church since it's inception. ((They)) control everything. Banks, religions, governments - you name it, they control it. Look to the history of Europe in the Middle Ages. All the power of the families was consolidated into the Papacy and The Families fought over the title of Pope for that reason.
The tip-top of the pyramid will be The Families. Right now The Families are aligned in a common cause - to destroy Humanity. They have put aside their infighting to achieve this goal. Had they achieved this end, they likely would have then fought it out amongst one another until only 3-4 Families remained and then would have called a truce among themselves.