  Let $G$ be an arbitrary finite group. The McKay conjecture asserts that $G$
and the normaliser $N_G (P)$ of a Sylow $p$-subgroup $P$ in $G$ have the same
number of characters of degree not divisible by $p$ (that is, of $p'$-degree).
We propose a new refinement of the McKay conjecture, which suggests that one
may choose a correspondence between the characters of $p'$-degree of $G$ and
$N_G (P)$ to be compatible with induction and restriction in a certain sense.
This refinement implies, in particular, a conjecture of Isaacs and Navarro. We
also state a corresponding refinement of the Brou\'e abelian defect group
conjecture. We verify the proposed conjectures in several special cases.
