On 2-maximal subgroups of finite groups

We distinguish 2-maximal and strictly 2-maximal subgroups of a finite group and give examples of soluble and simple groups in which every 2-maximal subgroup is strictly 2-maximal. Let M be a maximal subgroup of a group G. We prove that every maximal subgroup of M is strictly 2-maximal in G if M is normal in G or if G is p-soluble and vertical bar G : M vertical bar = p. We describe the structure of a finite group in which all 2-maximal subgroups are Hall subgroups. In particular, it has a Sylow tower and all its Sylow subgroups are elementary abelian.