H =H$.

Given a $G$-fibration $p:E\to B$ over $B$, the isotropy subgroup $G_b\leq G$ of a point $b\in B$ acts on the fiber space $F_b:=p^{-1} (b)$. So, $F_b$ is a $G_b$-space. Let us denote the set of isotropy subgroups of the $G$-action on $B$ by $\operatorname{Iso}(B)$.

Let $\{ F_H \}$ denote a family of $H$-spaces over all $H \in \operatorname{Iso}(B)$. If for every $b \in B$, the fiber space $F_b$ is $G_b$-homotopy equivalent to $F_{G_b}$, then $p:E \to B$ is said to have [*fiber type*]{} $\{ F_H \}$.

Note that in general a $G$-fibration does not have to have a fiber type, i.e., for $b_1, b_2 \in B$ with $G_{b_1}=G_{b_2}=H$, it may happen that $F_{b_1}$ and $F_{b_2}$ are not $H$-homotopy equivalent. But throughout the paper we only consider $G$-fibrations which have a fiber type. Observe that if $p: E \to B$ is a $G$-fibration such that $B^H$ is path connected for every $H \in \operatorname{Iso}(B)$, then $p$ has a fiber type since for every $b_1, b_2\in B^H$, the fiber spaces $F_{b_1}$ and $F_{b_2}$ are $H$-homotopy equivalent by a standard argument in homotopy theory. In our applications the $G$-fibrations that we construct will often satisfy this connectedness property.

If $p : 