$\def\Conj{\mathrm{Conj}}\def\Irrep{\mathrm{Irrep}}\def\Out{\mathrm{Out}}$Let $G$ be a finite group, let $\Conj(G)$ be the set of conjugacy classes of $G$, let $\Irrep(G)$ be the set of isomorphism classes of complex irreps of $G$ and let $\Out(G)$ be the outer automorphism group of $G$. Then $\Out(G)$ acts on both the sets $\Irrep(G)$ and $\Conj(G)$. (Since $\mathrm{Aut}(G)$ acts in an obvious way, and the inner automorphisms act trivially.)
The permutation representations $\mathbb{C}^{\Irrep(G)}$ and $\mathbb{C}^{\Conj(G)}$ are isomorphic (because evaluation of characters gives a perfect pairing between them, and both representations, being permutation representations, are self dual). Is there a group where the sets $\Conj(G)$ and $\Irrep(G)$ don't have an $\Out(G)$-equivariant bijection?
