I have a question about the construction of the so called "induced cover" introduced in Tamas Szamuely's "Galois Groups and Fundamental Groups" (see page 84):
We consider a group $G$ which contains a subgroup $H$. Futhermore we have a given cover $p: Y \to X$ of top space $X$ such that $H \subset Aut(Y \vert X)$. I the following excerpt we construct the so called induced cover $Ind_H ^G(Y)$ of $X$.
What I don't understand is why it follows that $Aut(Ind_H ^G(Y) \vert X) \cong G$ if the cover $p: Y \to X$ is Galois with group $H$, therefore $X \cong Y/H$.
My considerations: In the excerpt above is described how $G$ acts on $Ind_H ^G(Y)$, so we can denote $G \subset Aut(Ind_H ^G(Y) \vert X) $ as a subgroup of the automorphism group. Futhermore since the cover map $n:Ind_H ^G(Y) \to Y$ is given by $(g_iH, y ) \mapsto p(y)$ we can conclude that if $p$ Galois (so automorphism group acts transitive on the fibers $p^{-1}(x)$ for $x \in X$) then every $u \in Aut(Ind_H ^G(Y) \vert X) $ has following shape
$$(g_iH,y) \mapsto u(g_iH,y)= (u_1(g_iH,y), h(y))$$
for appropriate $h \in H = Aut(Y \vert X)$ and $u_1: Ind_H ^G(Y) \to G/H$.
And here comes the problem: how to show that $u$ in truth arises from a $g \in G$?
