McKay's conjecture for characters of $p'$-degree

For a finite group $G$ and a prime $p$, let

$$\operatorname{Irr}_{p'}(G):=\{\chi\in\operatorname{Irr}(G)\mid p\text{ does not divide }\chi(1)\}$$

be the set of irreducible complex characters of $G$ of degree prime to $p$. Let $P$ be a Sylow $p$-subgroup of $G$, and let $\operatorname{N}_G(P)$ be its normaliser. McKay's conjecture.

$$|\operatorname{Irr}_{p'}(G)|=|\operatorname{Irr}_{p'}(\operatorname{N}_G(P))|.$$

The conjecture asserts that fundamental information about the representation theory of $G$ is encoded in its Sylow normalisers. It was originally formulated for simple groups at $p=2$; it is known for groups of odd order and in several important families, but remains open in general.