Knightfall
World of Kulan DM
The idea that the Fraal might try to clean up the Earth would be an interesting twist, especially if they work towards reversing mutations. There are a lot of mutant groups that would be very opposed to this behavior. I could see mutated hitmen hunting fraal mercilessly.Another interesting aspect of this could be determining where the Grays/Fraals intentions would be (I believe the head of the Hoffman institute is actually a Gray/Fraal in disguise). They may try to help mankind clean up Earth. Possibly providing advanced equipment from their city ship that can help clean up radiation and possibly advanced medical technology to research reversing mutations. Hoffman Institute remnants could have the mission of making sure no new wars, world/planar plots, and new ecological disasters take place. You could even add in that maybe the Hoffman Institute had information that could have prevented the Apocalypse but were too late or unsuccessful, adding an element of guilt to their future activities.
The Hoffman Institute (HI) would be a major player in pure-strain human politics in a post-Dark Matter-Gamma World campaign. Depending on the GM, they could become a forward-thinking organization that includes mutants in their goals or an anti-mutant group that sees these upstarts as the greatest threat that humanity has ever faced.
Of course, that depends of a HI that is centralized. More than likely, the organization would splint into several groups with individual missions based on what you said above. (i.e. a splinter group dedicated to preventing wars, another dedicated to ecological problems, etc.) Perhaps many of these splinter groups become the Cryptic Alliances.
I also like the idea that the HI could have prevented the Final Wars but failed to act. The organization in the Gamma Age would work "really" hard to keep that information a secret. Otherwise, they could become the hunted by crazy-ass CAs that want to blame someone for the fall of civilization.