An n!-dimensional representation of the symmetric group S_{n+2} 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 $S_{n+1}$?

 A: 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 $S_{n}.$   
A: *

*Yes your series of representations looks (except for the first term - but there must be the law of small numbers lurking around) like the sign representation times the Whitehouse module, see, e.g. these slides by Richard Stanley. Basically, the Whitehouse module can be viewed as the respective component of the cyclic operad Lie. 

*The last sentence of the previous paragraph gives an operadchik's view on your second question. If your representation of $S_n$ appears as a component of arity $n$ of a certain operad, then the most "natural" situation in which it can be extended to $S_{n+1}$ is when your operad is cyclic or anticyclic (see, e.g. this paper of Getzler and Kapranov, and this paper of Chapoton for some flavour of the story).
