F_H\}$. If $B$ has trivial $H$-action, then the $k$-fold join is classified by a map $B \to B \operatorname{Aut}_H (\ast _k F_H)$. We would like to explain this map in terms of the classifying map of $p$. For this, observe that there is a monoid homomorphism $$\varphi : \operatorname{Aut}_H (F_H) \times \cdots \times \operatorname{Aut}_H (F_H)  \to  \operatorname{Aut}_H (\ast_k F_H)$$ defined by $$\varphi (a_1,\dots , a_k) (x_1 t_1, \dots ,x_k t_k)=(a_1(x_1) t_1 , \dots , a_k (x_k) t_k )$$ for every $x_1, \dots,  x_k \in F_H$ and $t_1, \dots , t_k \in [0,1]$ with $\sum_i t_i=1$. We have the following lemma:

\[lem:fiberjoins\] Let $H$ be a finite group, $F_H$ be a finite $H$-CW-complex, and $B$ be a CW-complex with trivial $H$-action. If $p: E\to B$ is a $H$-fibration with fiber type $F_H$ whose classifying map is $f: B \to B\operatorname{Aut}_H (F_H)$, then the classifying map of the $G$-fibration $\ast _k \, p$ is given by the composition $$\xymatrix@C=3pc{B \ar[r]^-{f\times \cdots \times f} & B\operatorname{Aut}_H(F_H)\times \cdots \times B\operatorname{Aut}_H (F_H) \ar[r]^-{B\varphi} & B \operatorname{Aut}_H (\ast _k F_H )}$$ where $B\varphi$ is the map induced from the monoid homomorphism $\varphi$ defined above.

Let $A=\prod _{i=1}^k \operatorname{Aut}_H  (F_H)$. By standard properties of homotopy pushout diagrams, 