Define the classifying space $BG$ of a well-pointed topological group $G$ as the fat realization of the nerve of $G$.
Let $G$, $H$ be well-pointed topological groups. Assume that there is a continuous group homomorphism $f:G\rightarrow H$ that induces a homotopy equivalence on the underlying topological spaces. It is claimed in an answer to this question that the induced map $BG\rightarrow BH$ is a homotopy equivalence. It is important here that we have a map between groups that simultaneously respects algebra and topology.
The question is basically what happens if we do not have such a map. Let $G$, $H$ be well-pointed topological groups. Assume that their underlying abstract groups are isomorphic and that their underlying topological spaces are homeomorphic. This by itself does not mean that $G$ and $H$ are isomorphic as topological groups (a pro-choice example is given by $p$-adic rationals for different $p$).
But what happens if we assume in addition that the classifying spaces of $G$ and $H$ are homotopy equivalent?
Note that a theorem of Notbohm says that two compact Lie groups are isomorphic as Lie groups iff their classifying spaces are homotopy equivalent so at least in some contexts, this question has a positive answer.
