Can one find non-isomorphic finite groups $G$ and $G'$ such that there is a homotopy equivalence $f:\Sigma^k BG \rightarrow \Sigma^k BG'$ for some $k$?
This would be impossible if we assumed $f$ was the iterated suspension of some pointed map $\varphi: BG \rightarrow BG'$. Indeed, homotoping rel base points if necessary, we can assume $\varphi= h_*$ for some homomorphism $h:G \rightarrow G'$. But this paper shows that if $h$ induces isomorphisms on integral homology, then it is an isomorphism of groups.
Examples involving infinite groups are not hard to find: $\Sigma B(\mathbb Z_{mn}) \simeq \Sigma B(\mathbb Z_m \ast \mathbb Z_n)$ for one, where $\ast$ is the free product and $m$ and $n$ are coprime. I'm not so sure about the finite case. Maybe it's straightforward?
I originally phrased this question in a very different, obscure way. I deleted my old question.