A generalization of Skiba's problem

Let $H/K$ be any chief factor of a finite group $G$, $p$ a prime, and $j$ a nonnegative integer. We say that $G\in\mathfrak{S}^{*}_{p^j}$ if every chief factor $H/K$ is a $p$-group or its $p$-part satisfies $|H/K|_{p}\leq p^j$. In this paper, we generalize Skiba's problem to the class $\mathfrak{S}^{*}_{p^j}$ of groups containing some non-solvable groups. We prove that $G\in\mathfrak{S}^{*}_{p^j}$ if and only if every maximal subgroup of a Sylow $p$-subgroup of $G$ has a subnormal supplement in $\mathfrak{S}^{*}_{p^j}$, where $p ＞ j$ or $p=j=2$.