Yes. The criterion is that $f:G\to H$ be an *$A_\infty$-map* with respect to the $A_\infty$-structures on $G$ and $H$. The definition is given in Section 4 of Stasheff's *Homotopy associativity of $H$-spaces, II* (Trans. Amer. Math. Soc. 108 (1963), 293–312), in particular see Theorem 4.5 there.

There was some related discussion at the MO question [Delooping maps between $H$-spaces][1].  


  [1]: https://mathoverflow.net/questions/71296/delooping-maps-between-h-spaces