  We describe the structure of finite groups with $\mathfrak{F}$-subnormal or
self-normalizing primary cyclic subgroups when $\mathfrak{F}$ is a
subgroup-closed saturate superradical formation containing all nilpotent
groups. We prove that groups with absolutely $\mathfrak{F}$-subnormal or
self-normalizing primary cyclic subgroups are soluble when $\mathfrak{F}$ is a
subgroup-closed saturate formation containing all nilpotent groups.
