Finite groups with systems of $K$-$\mathfrak{F}$-subnormal subgroups

Let $\mathfrak{F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_0\le A_1\le \cdots \le A_n=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$or$A_{i}/(A_{i-1}){A{i}} \in \mathfrak{F}$forall$i=1, \ldots , n$\mathrm{.}Aformation$\frak {F}$issaidtobe$K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattice formations. In particular, we prove the following Theorem A. Let $\mathfrak{F}$ be a hereditary $K$-lattice saturated formation containing all nilpotent groups. (i) If every $\mathfrak{F}$-critical subgroup $H$ of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$ with $H\mathrm{/}F(H)\in \mathfrak{F}$, then $G\mathrm{/}F(G)\in \mathfrak{F}$. (ii) If every Schmidt subgroup of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$, then $G\mathrm{/}{G}_{\mathfrak{F}}$ is abelian.