every map $c_g: H\to K$ induced by conjugation with $g\in G$, there exists a $\gamma \in U(n) $ such that the following diagram commutes $$\xymatrix{K \ar[d]^{c_g} \ar[r]^-{\alpha _K}
& U(n) \ar[d]^{ c_{\gamma }}  \\
H  \ar[r]^-{\alpha _H}
&  U(n) .}\\$$ Note that if $F_H$ is an $H$-space which is $H$-homotopy equivalent to ${\mathbb S}(V_H)$ for some compatible family of complex $H$-representation $V_H$, then $\{ F_H \}$ is a compatible family of $H$-spaces. So, the sphere bundle $p: S(E) \to B$ of a $G$-vector bundle is a spherical $G$-fibration with fiber type $\{ {\mathbb S}(V_H) \}$. Note also that for every $k\geq 1$, the fiber join $\ast _k {\mathbb S}(V_H)$ is $H$-homotopy equivalent to the $H$-space ${\mathbb S}(V_H ^{\oplus k})$ where $V_H ^{\oplus k}$ denotes the $k$-fold direct sum of $V_H$.

In Section \[sect:construction\], we construct $G$-fibrations with fiber types of the form $\{ {\mathbb S}(V_H)\}$. The following result is used in those constructions.

\[lem:joinswap\] Let $H$ be a finite group, $F_H$ be an $H$-space which is $H$-homotopy equivalent to ${\mathbb S}(V_H)$ for some complex $H$-representation $V_H$. Let $\gamma , \gamma ^1 : \operatorname{Aut}_H (F_H)\to \operatorname{Aut}_H (\ast _k F_H)$ be maps defined by $\gamma (a)= \varphi (a,a, \dots, a)$ and $\gamma 