Let $X$ be a projective variety, $D$ be an effective divisor, suppose there exists an invertible sheaf $L$ such that $L^n=\mathcal{O}(D)$, then we could obtain a cyclic cover $p:\tilde{X}\to X$ by taking n-th root out of $D$. If we denote $G$ be the deck transformation group of this covering, then $G=\mathbb{Z}_n$. I know the paper of Pardini who generalized the construction to ramified covering with $G$ an abelian group.
My question is:
Is there any algebraic construction of a ramified covering along an effective divisor $D$ with $G=S_n$, where $S_n$ is the symmetric group for n elements?
From topological point of view, the covering with $G=S_n$ will be equivalent of existence a representation $\rho:\pi_1(X\setminus D)\to S_n$ and the kernel of this map gives a n-fold covering. However, is there a building data for such kind of $\rho$ to exists?
