Groups with Formation Subnormal $2$-Maximal Subgroups

Groups with $\mathfrak X$-subnormal $2$-maximal subgroups are investigated for an arbitrary hereditary formation $\mathfrak X$. In such a group, all proper subgroups have nilpotent $\mathfrak X$-residuals. The cases in which $\mathfrak X=\mathfrak A_1\mathfrak F$ for some hereditary formation $\mathfrak F$ or $\mathfrak X$ is a solvable saturated formation are studied in more detail.