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)?