Strong 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. Strong inductive Giannelli–McKay condition. There should exist an $A$-equivariant bijection

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

such that $\Psi(\chi)(1)\leq\chi(1)$ and

$$(S\rtimes A_\chi,S,\chi)\geq_c(N_S(P)\rtimes A_\chi,N_S(P),\Psi(\chi))$$

for every $\chi\in\operatorname{Irr}_{p'}(S)$. This is a stronger version of the inductive condition, obtained in the cases treated in the paper. The source reports verification for exceptional Lie type in all primes and for groups of Lie type in defining characteristic, and consequently for all quasisimple groups when $p=2$, but does not establish the condition in full generality.