On Groups with Formational Subnormal Strictly 2-Maximal Subgroups

Let H be a subgroup of a finite group G. If G contains a maximal subgroup M such that H is a maximal subgroup of M, then H is called a 2-maximal subgroup of the group G. A subgroup U of G is said to be a strictly 2-maximal subgroup of G if U is a 2-maximal subgroup of G and U is not a 2-maximal subgroup in any proper subgroup of G. We investigate finite groups with X-subnormal strictly 2-maximal subgroups for any subgroup-closed formation X . In this group, any proper subgroup has a nilpotent X-residual. In more detail, we study the case where X = A 1 F for some subgroup-closed formation F and the case where X is a soluble saturated formation.