On finite groups in which every subgroup is either $\mathfrak{F}$-subnormal or $\mathfrak{F}$-abnormal

The structure of finite groups in which every proper subgroup is either $\mathfrak{F}$-subnormal or $\mathfrak{F}$-abnormal, where $\mathfrak{F}$ is a saturated hereditary formation with the Shemetkov property containing all nilpotent groups is studied. In particular, descriptions of these groups in the cases when $\mathfrak{F}$ is either the formation of all $p$-nilpotent groups or all $p$-decomposable groups were obtained.