Let $G$ be a finite group and $BG$ its classifying space. Let $[BG, BG]$ denote the set of self-maps of $BG$ up to homotopy equivalence. Automorphisms of $G$ give such self-maps, and inner automorphisms are homotopy equivalent to the identity. I am looking for an answer, hopefully with a reference if positive, to the following question:

> 
> Is it true that $\mathrm{Aut}(G)/\mathrm{Inn}(G) \equiv [BG, BG]$?
> 
> 
>