E \to B$ is a $G$-fibration with fiber type $\{ F_H \}$ such that for all $H \in \operatorname{Iso}(B)$ the fixed point space $B^H$ is connected, then the family $\{ F_H\}$ satisfies a certain compatibility condition. To see this, let $H , K \in \operatorname{Iso}(B)$ such that $K^g \leq H$ for some $g \in G$. Then, we have $gB^H \subseteq B^K$, so by the connectedness of $B^K$, we obtain that for every $b \in B^H$, the $K$-space $gF_b$ is $K$-homotopy equivalent to $F_K$. This means that the $K$-spaces $\operatorname{Res}_ K g^* F_H$ and $F_K$ are $K$-homotopy equivalent for all $H\in \operatorname{Iso}(B)$ where $\operatorname{Res}_K g^* F_H$ is the space $F_H$ which is considered as a $K$-space through the map $K \to H$ defined by $k \to g^{-1} kg$.

\[def:compatiblefamily\] Let ${\mathcal H}$ be a family of subgroups of $G$ closed under conjugation. A family of $H$-spaces $\{F_H\}$ over all $H \in {\mathcal H}$ is called a [*compatible family*]{} of $H$-spaces if for every $H, K \in {\mathcal H}$ with $K^g \leq H$ for some $g \in G$, the $K$-space $\operatorname{Res}_K g^* F_H$ is $K$-homotopy equivalent to $F_K$, where $\operatorname{Res}_K g^* F_H$ is the the space $F_H$ considered as a $K$-space through 