Inductive Giannelli–McKay condition for quasisimple groups

Let $S$ be a quasisimple group with cyclic center, let $p$ be a prime, let $P\in\operatorname{Syl}_p(S)$, and set $A=\operatorname{Aut}(S)_P$. For $\chi\in\operatorname{Irr}_{p'}(S)$, write $A_\chi$ for its stabilizer in $A$, and let $\geq_c$ denote the central-isomorphism relation on character triples. Inductive Giannelli–McKay condition. There should exist an $A$-stable subgroup $N_S(P)\leq M<S$ and an $A$-equivariant bijection

$$\Psi:\operatorname{Irr}_{p'}(S)\longrightarrow\operatorname{Irr}_{p'}(M)$$

with $\Psi(\chi)(1)\leq\chi(1)$ and

$$(S\rtimes A_\chi,S,\chi)\geq_c(M\rtimes A_\chi,M,\Psi(\chi))$$

for every $\chi\in\operatorname{Irr}_{p'}(S)$. This is a reduction condition intended to imply the global Giannelli–McKay conjecture from suitable verification on covering groups of simple groups. The paper presents it as a condition to be imposed in the reduction, and later verifies related cases, but no general resolution is stated.