Finite groups whose maximal subgroups have the Hall property

We study the structure of finite groups whose maximal subgroups have the Hall property. We prove that such a group $G$ has at most one non-Abelian composition factor, the solvable radical $S(G)$ admits a Sylow series, the action of $G$ on sections of this series is irreducible, the series is invariant with respect to this action, and the quotient group $G/S(G)$ is either trivial or isomorphic to $\mathrm{PSL}_2(7)$, $\mathrm{PSL}_2(11)$, or $\mathrm{PSL}_5(2)$. As a corollary, we show that every maximal subgroup of $G$ is complemented.