Example 1

Let F be a discrete set endowed with an operation of G, and let $X = E\times^GF$ be the associated covering of B. Let us denote by $\varphi : E\times F\rightarrow X$ the canonical B-morphism. It is a morphism of coverings. For every $\gamma \in \pi_1(B, b)$ and every $f\in F$, one then has

(5)

If one identifies F with $X_b$ by the bijective mapping $f\mapsto \varphi (x, f)$, it follows that the right operation $\pi_1(B, b)\rightarrow$ Aut(X$_b$)$^{\circ}$ is the composite of the homomorphism $h_{(E,x)}$, the antihomomorphism $g\mapsto g^{-1}$ of G into itself and the homomorphism $G\rightarrow$ Aut(F) deduced from the operation of G on F.