Obstructions for constructing equivariant fibrations

Let $G$ be a finite group and $\mathcal{H}$ be a family of subgroups of $G$ which is closed under conjugation and taking subgroups. Let $B$ be a $G$-$CW$-complex whose isotropy subgroups are in $\mathcal{H}$ and let $\mathcal{F}=\{F_H{\}}_{H\in \mathcal{H}}$ be a compatible family of $H$-spaces. A $G$-fibration over $B$ with fiber $$\mathcal{F}$= {F_H}{H \in $\mathcal{H}$}$isa$G$-equivariantfibration$p:E \rightarrow B$where$p^{-1}(b)$is$G_b$-homotopyequivalentto$F{G_b}$foreach$b \in B$. In this paper, we develop an obstruction theory for constructing $G$-fibrations with fiber $\mathcal{F}$ over a given $G$-$CW$-complex $B$. Constructing $G$-fibrations with a prescribed fiber $\mathcal{F}$ is an important step in the construction of free $G$-actions on finite $CW$-complexes which are homotopy equivalent to a product of spheres.