$S_n$ action on the power set of a set with $n - 1$ elements.

Question

Let $X_n = \{1, \ldots, n\}$. Let $S_n$ be the group of permutations of $X_n$. Let $S_{n - 1}$ lie inside $S_n$ such that $n$ remains fixed. For a set $M$ let $P(M)$ denote its power set. There is a natural action of $S_{n - 1}$ on $P(X_{n - 1})$. I'm looking for examples on $S_n$ actions of $P(X_{n - 1})$ which are compatible with the natural $S_{n - 1}$ action.