Let $k$ be a ring (resp. profinite ring), $G$ a group (resp. profinite group), and $k[G]$ the group algebra (resp. completed group algebra).
For any such $G$, we may associate to it the group of units $k[G]^\times$ of $k[G]$, and this association is clearly functorial. Has this functor been studied at all?
For example, given an exact sequence $$1\rightarrow G\rightarrow G'\rightarrow G''\rightarrow 1$$ we get a sequence $$1\rightarrow k[G]^\times/k^\times\rightarrow k[G']^\times/k^\times\rightarrow k[G'']^\times/k^\times\rightarrow 1$$ which is probably not exact, but at least the map $k[G]^\times/k^\times\rightarrow k[G']^\times/k^\times$ is injective, and so one might hope that the functor $G\mapsto k[G]^\times$ is left exact.
If it's left-exact, is this actually exact? If not, is anything known about its right-derived functors?
I suppose this is best posed under the assumption that $G$ is abelian, though I'm also very interested in the nonabelian case (or at least whatever still makes sense there in terms of cohomology)