If I have a compact connected Lie group $G$ and a (relatively nice) simply-connected topological abelian group $A$, when is it the case that a given $f\colon BG \to BA$ loops to a (continuous) *homomorphism* $G\to A$? This seems like it can't always be true, and also should be some kind of classical result, but I'm not sure where to look.



When does $BG \to BA$ loop to a homomorphism?
If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically equivalent to a homomorphism $\phi:G\to A$. This follows from the main result of 


*Scheffer, Wladimiro*, [**Maps between topological groups that are homotopic to homomorphisms**](http://dx.doi.org/10.2307/2038100), Proc. Am. Math. Soc. 33, 562-567 (1972). [ZBL0236.22008](https://zbmath.org/?q=an:0236.22008).


I'm not sure if this answers your question, however, which seems to be about when $f$ is in the image of the classifying space functor $B:\mathsf{TopGrp}\to\mathsf{Top}$.


