Is officially the diplomatic office, and sends Embassies to other nations and negotiate various matters with the Embassies of the Four Nations. It gets involved in all sorts of extracurricular activities.
The Faculty of Cartography is a branch of the University which receives almost all of its funding from Friendship. Ostensibly, it's there to make maps, and there's all sorts of interesting research going on there - after all, making maps which show Wallward/Truthward as well as North, South, East and West is a complicated process.
The most talented members of the Faculty of Cartography, however, and various agents of the Ministry of Friendship, operate the Treacherous Lands Exploration Initiative (or "the INITIATIVE"), which advertises for brave folk willing to go on prospecting expeditions beyond the Wall. Volunteers are brought up to the top of the Imperial Palace, taken to the Emperor's Gate, offered the blessing of a priest of the Intercessor if they so desire, and sent through the Gate. When they come back, they are greated with a large number of Watchdogs who keep their clockwork rifles trained on the explorers until it becomes clear that they're not shapeshifting horrors in human form. If they find something that's worth exploiting, they are asked to sign a nondisclosure and investment agreement: the deal is that they are to keep quiet about what they saw, but they get a nice percentage of any profits the government makes from exploiting their discovery.
Naturally, a dog is present at the debriefing to make sure they give a full account of their explorations.
Aside from this public work, the Initiative is also devoted to exploiting any resources discovered in the Treacherous lands, establishing colonies beyond the Wall, and entering into diplomatic relationships with those beyond.
A more paranoid and violent offshoot of the Initiative - which has been working more and more independently of late - is the Streetcleaners, the secret police responsible for keeping an eye out for anything nasty entering the True Lands, people smuggling things through alternate routes through the Wall, and that sort of thing.