  Let $\frak {F}$ 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
$$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1}
\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all
$i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-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 $H/F(H)\in {\mathfrak{F}}$, then
$G/F(G)\in {\mathfrak{F}}$.
  (ii) If every Schmidt subgroup of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$,
then $G/G_{\mathfrak{F}}$ is abelian.
