In p.91 of Szamuely's book "Galois groups and Fundamental groups", he defines the notion of 'induced $G$-cover' to extend the deck transformation group of covering. For given group $H\leq G$ and covering $p:Y\rightarrow X$ whose deck transformation group is isomorphic to $H$, the induced $G$-cover is defined by $$\text{Ind}_H^G(Y):=G/H\times Y$$ . And he claims that its deck transformation group is isomorphic to $G$ whenever $Y$ is Galois covering. However, in the case of $Y=X$, we could know that the deck transformation group of induced covering is $S_G$, symmetric group of $G$. I couldn't fully understand his statement about induced cover. Also, I couldn't believe that the deck transformation group of induced cover becomes exactly $G$. Is there anyone help me? Below picture is the construction in the book.
