I attempted the proof in Tamas Szamuely's book, Galois groups and fundamental groups. I am stuck in the last portion.
The cover $q \colon Z \to X$ is Galois if and only if $H$ is a normal subgroup of $G$, in which case $\operatorname{Aut(Z\mid X)} \cong G/H$.
I am not able to prove this part. Please suggest some hints and explanation for the above. Thanks.