Setup. Let $K$ be an algebraically closed field of characteistic zero, let $X/K$ be a smooth projective surface and let $Z \subset X$ be an integral curve which is nonsingular except for a finite set of cusps and simple double points, and let $U$ be the complement $X - Z$. Suppose that there exists a Galois cover (see here for definition) $U’ \to U$ with ''large'', non-abelian Galois group $G$; I do not have a specific definition for large here but for concreteness one could take $G \cong \text{GL}_r(\mathbb{F}_p)$ for some natural number $r\geq 2$ and prime $p$. Let $X’ \to X$ be the normalization in the ring of total fractions of $U’$, and let $f\colon Y \to X$ be the Galois closure of $X’ \to X$.
Questions. Can one say how many irreducible components $f^{-1}(Z)$ has in $Y$? If there are multiple components, can one describe how they intersect with each other?
Motivation. My motivation for this question comes from a construction of Faltings (see e.g., Section 1 of here) where he produced a family of singular irreducible curves $D$ in $\mathbb{P}^2$ such that $\mathbb{P}^2 - D$ has finitely many integral points. His proof roughly follows from defining suitable projections $f\colon X \to \mathbb{P}^2$ from a surface $X$ to $\mathbb{P}^2$ which have $D$ as a branch divisor and then studying the irreducible components of the ramification locus of the Galois closure $X' \to X \to \mathbb{P}^2$ of $X \to \mathbb{P}^2$ (the precise description of these components appears in part (v) of the above reference).
Thank you in advance!
