A new generalization of the McKay conjecture for $p$-solvable groups

Let $P$ be a Sylow $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime. Using a normal $p$-series $\mathcal{N}$ of $G$, we introduce the notion of $(\mathcal{N},p)$-stable characters and prove that $G$ and ${\mathbf{N}}_G(P)$ have equal numbers of such characters, which gives a new generalization of the McKay conjecture for $p$-solvable groups. Also, we establish a canonical bijection between these characters in the case where $G$ has odd order. Our proofs depend heavily on the theory of self-stabilizing pairs founded by M. L. Lewis, as well as some results of $\pi$-special characters due to I. M. Isaacs.