TITLE: An n!-dimensional representation of the symmetric group S_{n+2}
QUESTION [22 upvotes]: I have come across a sequence of representations $V_n$ of the symmetric group $S_{n+2}$ which has the property that restricting the action $S_n \subset S_{n+2}$ gives the regular representation:
$$ Res^{S_{n+2}}_{S_n} V_n = \mathbb{Q}S_n. $$
In other words, there is some natural way to give the regular rep of $S_n$ an action of $S_{n+2}$.  This (to me) is surprising, but I imagine this has already been observed.  
For concreteness, here are the first few terms in the sequence, written as a sum of partitions (using the usual indexing of representations of $S_{n+2}$ by partitions of $n+2$):
[2]
[3]
[2,2]
[3,1,1]
[3,3]+[2,2,1,1]+[4,1,1]
...

I have two questions:

Is there already work on unrestrictions of the regular representation of a symmetric group?  Is my particular sequence of representations $V_n$ well-known?
In general, are there circumstances under which a representation of $S_n$ has a canonical way to extend the action to $S_{n+1}$?

REPLY [8 votes]: When $n+2$ is a prime $p,$ it is relatively easy to construct such a representation. For then the symmetric group $S_{p}$ contains a Frobenius group $F$ of order $p(p-1).$ Every element of $F$ either has order $p$ or has order dividing $p-1.$ Furthermore, each non-identity element of $F$ of order dividing $p-1$ has exactly one fixed point in the natural permutation action of $S_{p}.$ Consequently, if we induce the trivial module for $F$ to $S_{p},$ and restrict that module back to the natural copy of $S_{p-2},$ then we obtain the regular module for $S_{p-2}$ (using for example, Mackey's formula and the fact that $[S_{p}:F] = (p-2)!,$ and noting that $F^{x} \cap S_{p-2} = 1$ for all $x \in S_{p}).$ However, I do not see offhand an easy way to generalize this argument when $n+2$ is not prime
Note also that a rather simpler argument using the cyclic subgroup $X$ generated by an $n+1$ cycle shows that (for general $n$), the permutation module afforded by the action of $S_{n+1}$ on the (say right) cosets of $X$ restricts to the regular module of $S_{n}.$