Finite groups with generalized subnormal $\mathfrak{F}$-critical subgroups

Let $G$ be a finite group and let $A$ be a subgroup of $G$. Let $A_{\operatorname{sn}G}$ be the subgroup of $A$ generated by all subnormal subgroups of $G$ contained in $A$, and let $A^{\operatorname{sn}G}$ be the intersection of all subnormal subgroups of $G$ containing $A$. Let $N\leqslant G$. Then we say that $A$ is $N$-subnormal in $G$ if $N$ avoids every composition factor $H/K$ of $G$ between $A_{\operatorname{sn}G}$ and $A^{\operatorname{sn}G}$, i.e., $N\cap H= N\cap K$. In this paper, we give applications of $N$-subnormality to the theory of groups with given $\mathfrak{F}$-critical subgroups. In particular, using this notion, we give new characterizations of finite solvable groups, metanilpotent groups, and groups with nilpotent derived subgroup $G'$.