over the trivial $H$-space ${\mathbb S}^n$ where $n\geq 2$. Suppose that the fiber type $F_H$ of $p$ is $H$-homotopy equivalent to ${\mathbb S}(V_H)$ for some complex $H$-representation $V_H$. If  $\pi _{n-1} ( \operatorname{Aut}_H (F_H ))$ is a finite group of order $N$, then $\ast _N p$ is $H$-fiber homotopy equivalent to the trivial fibration.

By Theorem \[thm:classification\], the $H$-fibration $p$ is classified by the homotopy class of a map $f: {\mathbb S}^n \to B\operatorname{Aut}_H (F_H)$. Since $n\geq 2$, this map lifts to map $\widetilde f : S^n \to B\operatorname{Aut}^{I}_H (F_H)$, so we can assume that the $H$-fibration $p$ is an homotopy orientable fibration . Since $B\operatorname{Aut}^I _H (F_H)$ is simply connected, we have $$[S^n, B\operatorname{Aut}^I _H (F_H )]\cong \pi _n (B\operatorname{Aut}^I _H (F_H))\cong \pi_{n-1} (\operatorname{Aut}_H (F_H)).$$ So, $p$ is classified by a homotopy class $\alpha\in \pi _{n-1} (\operatorname{Aut}_H (F_H))$. By a slightly modified version of Lemma \[lem:fiberjoins\], it is easy to see that the fiber join $\ast _N p$ is classified by $\gamma _* (\alpha)$ where $\gamma: \operatorname{Aut}_H (F_H) \to \operatorname{Aut}_H ( \ast _H F_H )$ is the map defined by $\gamma (a)=\varphi (a,\dots, a)$. By Lemma \[lem:joinswap\], we have $$\gamma _* (\alpha)=N\gamma _* ^1 (\alpha)=\gamma _* ^1 (N \alpha 