Let $f\colon X\to Y$ be an H-map between homotopy associative H-spaces in the homotopy category of based CW-complexes. It's well-known that there is an induced H-map from the homotopy fibre $F$ of $f$ to $X$; that is, there is a homotopy fibration of H-spaces and H-maps: $$\Omega Y\xrightarrow{\partial} F\xrightarrow{i} X\xrightarrow{f}Y.$$ Suppose that $F$ is also homotopy associative and let $\mathcal{B}F$ be the classifying space of $F$. My question is: Under what conditions that there is an induced map $\mathcal{B}\partial\colon Y\to\mathcal{B}F$ such that the fibration extends from the right as $$\cdots\to F\xrightarrow{i} X\xrightarrow{f}Y\xrightarrow{\mathcal{B}\partial}\mathcal{B}F.$$
I know it is always true if $X,Y$ are topological groups, while from Page 31 of Stasheff's book "H-spaces from a homotopy point of view" I read that an H-map doesn't necessarily induces a map between the classifying spaces. I didn't find the answers to the question in Stasheff's book.