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 $S_{n+1}$?