What subgroups of $\operatorname{Iso}[X,X]$ are realized as holonomy groups of a fibration? By $\operatorname{Iso}[X,X]$ I mean the self homotopy equivalences of $X$.
What I mean by holonomy group of a fibration $F \rightarrow E \rightarrow B$ is the image of $\pi_1(B,b_0)$ in $\operatorname{Iso}[F,F]$ by the map taking a path in $B$ to the associated homotopy equivalence.
In particular, I am interested in the case where $F$ is an H-space (or possibly more).
For motivation, I am interested if you can study self homotopy equivalences by way of constructing particular fibrations, and this seems like the natural question to start with.
Some thoughts: Given a discrete group $G$, standard covering space theory tells us that we can realize all translations by $G$ as the holonomy of the universal G-bundle. So, perhaps it is best to restrict the question to maps that respect the H-space structure of $F$.