the map $K \to H$ defined by conjugation $k \to g^{-1} k g$.

The main aim of this section is to introduce some tools for construction of $G$-fibrations with fiber type $\{ F_H\}$ for a given compatible family $\{ F_H\}$. We first introduce some more terminology:

Given two $G$-fibrations $p_1:E_1 \to B$ and $p_2: E_2 \to B$ over the same $G$-space $B$, a $G$-map $f: E_1\to E_2$ is called a fiber preserving map if it satisfies $p_2\circ f =p_1$. Two fiber preserving $G$-maps $f, f': E_1\to E_2$ are said to be $G$-fiber homotopic if there is a $G$-map $H: E_1\times I \to E_2$ which is fiber preserving at each $t\in I$ such that $H(x, 0)=f(x)$ and $H(x,1)=f'(x)$ for all $x\in E_1$. We say two $G$-fibrations $p_1 :E_1 \to B$ and $p_2: E_2 \to B$ are $G$-fiber homotopy equivalent if there are fiber preserving $G$-maps $f_1:E_1\to E_2$ and $f_2: E_2 \to E_1$ such that $f_1\circ f_2$ and $f_2 \circ f_1$ are $G$-fiber homotopic to identity maps.

For an $H$-space $F_H$, let $\operatorname{Aut}_H (F_H)$ denote the topological monoid of self $H$-homotopy equivalences of $F_H$. Note that $\operatorname{Aut}_H (F_H)$ is not a connected space in general but it is easy to show that 