Let $H$ be a finite $p$-group, and let $G$ be a compact connected Lie group. Then 
it is well-known that $[BH,BG]\cong Rep(H,G)$, where $BH$ and $BG$ are classifying spaces and $Rep(H,G)$ is the set of representations of $H$ in $G$. What happens if we assume that $H$ is any finite group -- do we also have $[BH,BG]\cong Rep(H,G)$?



Representation of finite groups in a compact Lie group
In "Maps from $B\pi$ into $X$" Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 153, 117–127., Wojtkowiak proves that the natural map 
$Rep(H,G)\rightarrow [BH,BG]$ is not surjective when $H=\Sigma \_3$, $G=U(2)$.


