Let $H$ be a finite group acting on a group $G$.  Are there general conditions one can place on $G$ so that the set $H^1(H, G)$ is guaranteed to be finite?  For instance, does finiteness hold if $G$ is finitely presented, or if $K(G, 1)$ is a finite CW complex?

**Added:**  According to Reid Barton's answer [here](http://mathoverflow.net/questions/2390/why-does-non-abelian-group-cohomology-exist), $H^1(H, G)$ is identified with $\pi_0(X^{hH})$, where $X$ is the space $K(G, 1)$ and the $hH$ denotes homotopy invariants.  Thus it would be enough to know the following:  if a finite group $H$ acts on a finite CW complex $X$, is $\pi_0(X^{hH})$ finite?