14.30 (1999)

Let $\operatorname{lFit} \mathfrak{X}$ be the local Fitting class generated by a set of groups $\mathfrak{X}$ and let $\Psi(G)$ be the smallest normal subgroup of a finite group $G$ such that $\operatorname{lFit}(\Psi(G) \cap M) = \operatorname{lFit} M$ for every $M \triangleleft\triangleleft\ G$ (K. Doerk, P. Hauck, Arch. Math., 35, no. 3 (1980), 218–227). We say that a Fitting class $\mathfrak{F}$ is saturated if $\Psi(G) \in \mathfrak{F}$ implies that $G \in \mathfrak{F}$. Is it true that every non-empty soluble saturated Fitting class is local?