Outer automorphism action on representations of $S_6$

Question

Let $S_6$ be the symmetric group on 6 letters and let $\alpha \colon S_6 \to S_6$ be an outer automorphism (note that $S_6$ is the only permutation group that has an outer automorphism and that $\mathrm{Out}(S_6) \cong \mathbb{Z}/2\mathbb{Z}$). For any irreducible representation $\rho \colon S_6 \to \mathrm{GL}(V)$ of $S_6$, the composition $\rho \circ \alpha \colon S_6 \to \mathrm{GL}(V)$ is also an irreducible representation.

Question: Is there a reference describing the action of this operation ($\rho \mapsto \rho \circ \alpha$) on the set of all irreducible representations of $S_6$ over $\mathbb{C}$ (that is on the set of partitions of 6)?

Answer 1

First, notice that this operation preserves the dimension of the representation, this already considerably restricts things: the dimensions of the irreps. are 1,1,5,5,5,5,9,9,10,10,16. The trivial rep. is fixed, thus the sign rep. must also be. The 16-dimensional rep. $V_{(3,2,1)}$ is also fixed.

Next, since $\alpha$ maps conjugacy classes to conjugacy classes, it preserves the multiset of character values of $\rho$. Checking this against a character table shows that the 9-dimensional reps. are also all fixed. This does allow for the possibility that the 5-dimensional rep. $V_{(5,1)}$ is sent to $V_{(2,2,2)}$ and its conjugate $V_{(2,1,1,1,1)}$ is sent to $V_{(3,3)}$. You can see that one of these pairs is swapped if and only if the other is by tensoring with the sign representation. The 10-dimensional irreps. $V_{(4,1,1)}$ and $V_{(3,1,1,1)}$ may also be swapped.

EDIT: After I have written this, Mark Wildon has linked to a question which subsumes this one. His answer there, using an explicit description of $\alpha$'s action on conjugacy classes, is that all of the possible swaps I identified do in fact occur.