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