If you consider an exact sequence of groups, it's a fibration of topological spaces, so you get a homotopy long exact sequence. So that makes me think $\pi_n$ behaves like a derived functor from topological abelian groups to abelian groups. So there should be a spectral sequence whose first page is $\pi_m (G_n)$ and whose second page is $\pi_m$ of cohomology of $G_n$.
Because the cohomology of the first page is the second page, I think this implies that cohomology of $\pi_m (G_n)$ is just equal to $\pi_m$ of cohomology of $G_n$ (if this derived functor heuristic is correct, and I'm not very good at spectral sequences so it might be false regardless).