we observe that the fibration $\ast _k \, p$ is the pullback fibration of the fibration $$q: EA \times _A (\ast _k F_H ) \to BA$$ via the map $\prod _i  f: B \to BA$. Note that $A$ acts on $\ast_k F_H$ via the monoid homomorphism $\varphi$, so the classifying map of $q$ is $B\varphi$. This completes the proof.

A special case of a $G$-fibration is a $G$-fiber bundle over a $G$-CW-complex. More specifically, if $\xi : E\to B$ is a complex $G$-vector bundle over a $G$-CW-complex $B$, then the sphere bundle $p: S(E) \to B$ of this vector bundle is a spherical $G$-fibration. Note that for every $b \in B$, the fiber space $p^{-1} (b)$ is a $G_b$-space which is homeomorphic to ${\mathbb S}(V_{G_b})$ where $V_{G_b}$ denotes vector space $\xi ^{-1}(b)$ with the induced $G_b$-action.

Note that when $B^H$ is path connected for all $H \in \operatorname{Iso}(B)$, the family of complex representations $\{ V_H\}$ defined over all $H \in \operatorname{Iso}(B)$ is a compatible family. The compatibility of a family of representations is defined in the following way: A family of representations $\alpha_H :H \to U(n)$ over $H \in {\mathcal H}$ is called a [*compatible family of representations*]{} if for 