The Division

An earlier incarnation of the Doctor in their "Timeless Child era" was contacted, tested and recruited into the Division by Solpado. Where the policy of the Time Lords was of "strict non-intervention in other worlds and times", the purpose of the Division was to intervene in times when it was necessary. Officially the Division did not exist nor had operatives. Its history was mostly redacted from the Matrix. Eventually, they thanked the Doctor for their service, then wiped their memory. In the fictional story of Brendan Bildbriain in the Matrix, one of the clues to the truth was that the carriage clock he was given upon his retirement was dedicated for services to the Division instead of the Garda. (TV: The Timeless Children)

The Gallifreyan Gat and her recruited troop of Judoon, led by Pol-Kon-Don, also worked for the Division. When the Doctor tried to escape, their job was to capture her and punish her for running away. (TV: Fugitive of the Judoon)