19.38 (2018)

Suppose that $H$ is a subgroup of a finite soluble group $G$ that covers all Frattini chief factors of $G$ and avoids all complemented chief factors of $G$. Is it true that there are elements $x, y \in G$ such that $H \cap H^x \cap H^y = H_G$, where $H_G$ is the largest normal subgroup of $G$ contained in $H$? This is true if $H$ is a prefrattini subgroup of $G$.