(Some of the notational choices I"m about to make might be iffy; I'm happy to take suggestions for improvements.)

Let $G$ be a (discrete) group. Think of it as an object in the $2$-category of small categories; then it has an automorphism $2$-group $\text{Aut}(G)$, and this automorphism $2$-group has a classifying space $B \text{Aut}(G)$ given, for example, by taking the geometric realization of the nerve. On the other hand, let $X$ be a space with (weak) homotopy type $BG$. Under mild conditions $\text{Aut}(X)$, the group of homeomorphisms from $X$ to itself, is a topological group, which also has a classifying space $B \text{Aut}(X)$. (**Edit:** The comments below strongly suggest I should instead be talking about the topological monoid of homotopy equivalences from $X$ to itself, so let's do that instead.)

Are these classifying spaces (weakly) homotopy equivalent?

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