Finite groups with $\mathscr F$-subnormal conditions

Let $\mathscr F$ be a subgroup-closed saturated formation. A finite group $G$ is called an $\mathscr F_{pc}$-group provided that each subgroup $X$ of $G$ is $\mathscr F$-subabnormal in the $\mathscr F$-subnormal closure of $X$ in $G$. Let $\mathscr F_{pc}$ be the class of all $\mathscr F_{pc}$-groups. We study some properties of $\mathscr F_{pc}$-groups and describe the structure of $\mathscr F_{pc}$-groups when $\mathscr F$ is the class of all soluble $\pi$-closed groups, where $\pi$ is a given nonempty set of prime numbers.