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At first, Vertin's team is mostly used to exterminate leftover traces of Manus Vindictae. They hunt them down all over the globe.
Vertin still recruits people to both the Foundation and her department along the way. That is one of the Foundation's main goals after all. However, she has all the "unhinged" or "dangerous" arcanists under her care since the Foundation doesn't know what to do with them, but doesn't want them causing trouble.
Then, the Timekeeper Department moves more toward problem solving. They are sent to investigate rumors and situations when the Foundation doesn't want to use their own manpower. They will also deal with Critters and Arcane shenanigans. Because Vertin's Department is so diverse, they also do humanitarian efforts. Druvis can help a poor village with failing crop yields. Lilya can deliver medication and other resources in emergencies. There are arcanists who can track down bad people and also find people who are lost. These are just a few examples but I'm sure there are many more. This also makes the Foundation look great and boosts their reputation.
However, as time goes on the problems Vertin's Department are sent to face become increasingly dangerous for them to handle alone. Things that the Foundation should be using their own people for or at least help with.
It's the same issue as before: the Foundation is afraid Vertin will turn against them. She now has the means to do so. The Foundation may have more people and resources, but Vertin's people have experience, pure power, and unwavering loyalty. Vertin has tried to assure them that she just wants to live in peace, but they are still terrified of the possibility.
So they've started sending Vertin's people on more dangerous and sometimes even unnecessary missions hoping to thin her numbers out. Her arcanists are too dangerous for society anyway. They will die as martyrs. At the same time, The Foundation hopes to make Vertin's people lose faith in their leader. They want to create internal conflicts and want to instill someone more loyal to the Foundation as the head of the Department.
But they can't find anyone who can reign in the Arcanists like Vertin. They've tried bribing and making offers to her people directly but it always blows up in their face.
Vertin's personal connections with her people seem irrational and far too emotional. Her people will follow her into hell but the problem is they keep coming back. The Foundation is forced to acknowledge that they have no control over the situation anymore. This insight causes more tension in the Foundation.
They thought they trapped Vertin, but it turns out their the ones trapped with her. They can't get rid of her or else her people will run loose. They can't replace her because her department won't accept anyone else.
Vertin still does the missions, but demands proper resources. She still leads her people to help others and save arcanists seeking refuge. However, whenever the Foundation gets too pushy or unreasonable during a meeting, all she has to do is place her Suitcase on the table. The unsaid threat is more than enough to shut them up.
Constantine is actually on Vertin's side in all of this. She's tired of all the cowards yapping. As much as she hates to admit it, she's seen Vertin grow into the force she is today. She and Vertin openly argue about things, which makes other people on edge but it's normal for them. They have an understanding: Vertin will help the Foundation, and the Foundation will help her. That's all there is to it but only Constantine seems to get that.
On that note, Vertin is one of Constantine's greatest achievements. Constantine is confident in her abilities and judgement, which in a way translates to an odd form of trust in Vertin. Maybe not in Vertin herself per say, but trust in the way Vertin will act.
Madam Z thinks Vertin has grown on Constantine, but Constantine looked absolutely disgusted by the assumption.