Socle over solvable radical

Definition
Suppose $$G$$ is a finite group. The socle over solvable radical of $$G$$, denoted $$\operatorname{Soc}^*(G)$$, is defined as the unique subgroup $$H$$ of $$G$$ such that $$H$$ contains the defining ingredient::solvable radical $$\operatorname{Rad}(G)$$ of $$G$$ and the quotient group $$H/\operatorname{Rad}(G)$$ is the defining ingredient::socle of the quotient group $$G/\operatorname{Rad}(G)$$.

Facts
If $$H$$ is the socle over solvable radical of $$G$$, then $$\operatorname{Rad}(H) = \operatorname{Rad}(G)$$ and $$H/\operatorname{Rad}(H)$$ is a direct product of simple non-abelian groups.