Papal Mainframe

The Papal Mainframe was an ecclesiastical leader whose authority gave Colonel Manton the ability to lower the hoods on headless monks — an act that was otherwise a level 1 heresy. At the time Manton took the action at Demons Run, the Mainframe was either actually female or commonly referred to using feminine pronouns. (DW: A Good Man Goes to War)

Behind the scenes
In behind the scenes sources, the Papal Mainframe was said to be the leader of the |Headless Monks (REF: Doctor Who: The Encyclopedia) or the Church. (REF: The Brilliant Book 2012)