Let $X:=\mathbb{C}^n$, and let the symmetric group $S_n$ act by permutation of coordinates in the obvious way; let $X_n:=X/S_n$ be the quotient by the group action. Now, $X_n\simeq \mathbb{C}^n$, so we get a map $\pi: X\to X_n$ with finite fibers. Away from the discriminant $\Delta:=\prod_{i\lt j}(x_i-x_j)$, $\pi$ is a covering map $X\setminus\Delta\to X_n\setminus\pi(\Delta)$, and this map has monodromy group $S_n$. In similar fashion, we can let the subgroup $S_k$ act on the first $k$ coordinates, and let $X_k$ be the resulting quotient; again away from the discriminant the map $X\to X_k$ is a covering map and has monodromy group $S_k$. (We could of course generalise to $S_\lambda$ for a partition $\lambda$ of $n$, but let's keep it simple.)

Now, $\pi:X\to X_n$ factors through any of the $X_k$, so we have maps $X_k\to X_n$, and in particular $X_n\to X_{n+1}$, and away from the branching locus they are also covering maps.

Question: What are the monodromy groups of these covering maps?

This seems like the sort of thing someone might have worked out long ago, is anything known about this?