When is $BG$ a topological group? Let $G$ be a topological group, then it has a classifying space $BG$. 

When is $BG$ a topological group? 

My motivation for asking this question is that I was thinking about the $B$-analogue of my previous question and realised it's not clear whether $B^kG$ is defined for $k > 1$.
One case where I know it is true is when $G$ is an Eilenberg-Maclane space, i.e. $G = K(H, n)$. First of all, they are topological groups (see here) and by considering the long exact sequence in homotopy associated to the fibre bundle $G \to EG \to BG$, we see that $BG \cong K(H, n+1)$ and is therefore a topological group. It is also true when $G$ is a discrete group for similar reasons.

Up to homotopy, the answer is if and only if $G$ has the additional structure of an $E_2$ space. (This is exactly the structure that a double loop space has; we need this since a topological group has a classifying space, so if $BG$ has a classifying space then $G$ is a double loop space.)
This is a certain higher categorical version of abelianness. If $G$ is discrete, it must be abelian (by the Eckmann-Hilton argument, which is really about $E_2$ structures), but there are interesting examples beyond abelian or topological abelian groups, such as (again, up to homotopy) the stable unitary group $U$. 
The easiest case past the discrete case is the case that $G$ is a groupoid; then $E_2$ means $G$ is a braided monoidal groupoid where every object is invertible. These have a cohomological classification. 
