Let $H$ and $G$ be topological groups and denote by $BH$ and $BG$ their classifying spaces.
If $$f\colon H\rightarrow G$$ is a continuous group homomorphism, we get an induced map of spaces $$Bf\colon BH\rightarrow BG.$$
Now suppose $f\colon H\rightarrow G$ is only a group homomorphism up to homotopy, i.e. the diagram $$\require{AMScd} \begin{CD} H\times H @>{m_H}>> H;\\ @V{f\times f}VV @V{f}VV \\ G\times G @>{m_G}>> G; \end{CD}$$ commutes up to homotopy.
Do we get an induced map $Bf\colon BG\rightarrow BH$. If this depends on the model of $B(\_)$ I would be interested in the question whether their exists a model such that this works out.
Edit: I am willing to accept choices, as long as the homotopy class of $Bf$ is independent of those.