Here is the $1$-type case. I assume all spaces are of the homotopy type of CW. Let me write $haut(X)$ (resp. $haut_*(X)$) for the monoid of self-equivalences (resp. pointed ones) to avoid posible confusion with the group-theoretic notation. These spaces have the correct homotopy type by our assumption.


From [*All Groups are Outer Automorphism Groups of Simple Groups*](http://dx.doi.org/10.1112/S0024610701002484) by Droste,
Giraudet and Göbel, for every discrete group $G$ we can write $G\cong out(S)$ where $S$ is simple. Recall the exact sequence $0\to Z(S)\to S\to aut(S)\to out(S)\to 0$. If $S$ is abelian we are done, so we can assume $S$ has trivial center.

Looping down the universal fibration with fiber $BS$ we have the homotopy principal fiber sequence
$S\to haut_*(BS)\to haut(BS)$ and thus $haut_*(BS)//S\simeq haut(BS)$. But $haut_*(BS)\simeq aut(S)$ and $S\to aut(S)$ is injective so the homotopy quotient $haut_*(BS)//S$ is the (ordinary) quotient $aut(S)/S\cong out(S)\cong G$.
Hence, $G\simeq haut(BS)$.