Permutation kernel

Definition
Suppose $$G$$ is a finite group, $$R = \operatorname{Rad}(G)$$ is its defining ingredient::solvable radical, and $$S = \operatorname{Soc}^*(G)$$ is its defining ingredient::socle over solvable radical, i.e., $$S/R$$ is the defining ingredient::socle of $$G/R$$. $$S/R$$ can be expressed uniquely as a direct product of simple non-abelian groups.

The permutation kernel of $$G$$, denoted $$\operatorname{PKer}(G)$$, is the kernel of the action of $$G$$ on these factors induced by the action of $$G$$ on $$S/R$$ by conjugation. $$\operatorname{PKer}(G)$$ contains the socle over solvable radical $$S$$.

The permutation kernel is part of the Babai-Beals filtration of $$G$$.