In Lawrence-Venkatesh, they tried to descend their construction of universal branched $G$-cover $Z^\circ\to Y^2-\Delta$ in Lemma 7.4. I have several questions about the proof.
They said the commuting action $\Gamma\times G$ on $S(\Gamma,c,G)$ defines a cover over $Y-\{y_0\}$, which is precisely the disjoint union of all $G$-covers branched at $y$, I could not understand this correspondence.
He said the action of $\Gamma$ on $S(\Gamma,c,G)/G$ is trivial, maybe this follows from easy calculation. But why then the action of $\tilde{\Gamma}$ factors through $\pi_1(Y_K,y_0)$.
