Papal Mainframe

She gave Colonel Manton of the Church, permission to lower the hoods on the Monks — an act that was otherwise a level one heresy.

The Mainframe was referred to with feminine pronouns and had a female voice when talking to one Cleric. (DW: A Good Man Goes to War)

Behind the scenes
The Papal Mainframe was the ecclesiastical leader of the Headless Monks. (REF: Doctor Who: The Encyclopedia) A Papal Mainframe, possibly the same one, also gave commands to the Church. (REF: The Brilliant Book 2012) Being called a "mainframe", she was likely an artificial intelligence.

''The two reference books, published in the same year, disagree on whether the Mainframe oversees the Monks or the Church. The episode is ambigious. She could be the overseer of multiple religions.''