Suppose given a map $f:X\to Y$ with homotopy fibre a loop space $\Omega Z$. Then there is a homotopy equivalence $\Omega Z\simeq G$ for some topological group $G$. Does there exist a principal $G$-bundle $E\to Y$ with a fibrewise homotopy equivalence $X\simeq E$? If not, are there any calculable obstructions to that? Maybe there are some additional structures on $f$ ensuring existence of such equivalence?
I remember having seen somewhere that the homotopy categories of $G$-spaces and spaces over the classifying space $BG$ are equivalent, but I don't quite see whether this answers my question. Presumably one should find a map $Y\to Z$ with fibre $X$ or something like that, but I don't know whether there are any obstructions to this, or whether these obstructions are of the same nature as the ones I mentioned before.