  Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay
conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup
have equal numbers of irreducible characters with degrees not divisible by $p$.
The Alperin-McKay conjecture is a version of this as applied to individual
Brauer $p$-blocks of $G$. We offer evidence that perhaps much stronger forms of
both of these conjectures are true.
