Apologies for the title.
Miller's theorem (formerly Sullivan's conjecture) gives that for a finite group $G$ and a finite dimensional connected CW complex $X$, the based mapping space $\operatorname{Map}_*(BG,X)$ is weakly contractible. In particular, by consideration of $\pi_0$ we see that every map $BG\to X$ is null-homotopic.
A generalization of Sullivan's conjecture to $G$-CW complexes $X$ with non-trivial action was proved Dwyer-Miller-Neisendorfer, Lannes and Carlsson. It states that if $G$ is a $p$-group and $X$ is a finite dimensional $G$-CW complex, then the natural map $$ X^G \to \operatorname{Map}_G(EG,X) $$ from the fixed points to the homotopy fixed points is a weak equivalence after $p$-completion. In particular, this map induces an isomorphism on mod $p$ homology and consequently induces a bijection $$ \pi_0(X^G) \cong \pi_0(\operatorname{Map}_G(EG,X)) $$ on path components. This could be viewed as a weakened generalized Sullivan's conjecture.
Does the weakened Sullivan conjecture hold for all finite groups? That is, let $G$ be a finite group and $X$ a finite dimensional $G$-CW complex. Is it true that $$ \pi_0(X^G)\cong\pi_0(\operatorname{Map}_G(EG,X))? $$ If not, what extra hypothesis could be put on $X$ so that this holds?
