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 $f\colon BG \to BA$ deloops 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.

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.