On residuals of finite groups

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size soluble residual is bounded from below by certain power order, inequality sharp. Inspired this result some arguments \cite{DHKL}, we establish following generalisation: if $\mathfrak{X}$ subgroup-closed formation full characteristic which does not contain all groups $\overline{\mathfrak{X}}$ extension-closure $\mathfrak{X}$, then there exists an (optimal) constant $\gamma$ depending only on such that, for non-trivial $G$ $\mathfrak{X}$-radical, \begin{equation} \left\lvert G^{\overline{\mathfrak{X}}}\right\rvert \,>\, \vert G\vert^\gamma, \end{equation} where $G^{\overline{\mathfrak{X}}}$ ${\overline{\mathfrak{X}}}$-residual $G$. When $\mathfrak{X} = \mathfrak{N}$, class nilpotent groups, it follows $\overline{\mathfrak{X}} \mathfrak{S}$, thus recover original Lev. In last section our paper, building J.\,G. Thompson's classification minimal simple exhibit family formations $\mathfrak{S} \subset \overline{\mathfrak{X}} \mathfrak{E}$, providing applications main beyond reach \cite[Thm.~C]{DHKL}.