On $\mathrm{K}\mathfrak F$-subnormality and submodularity in a finite group

Let $\mathfrak F$ be a formation and let $G$ be a group. A subgroup $H$ of $G$ is $\mathrm{K}\mathfrak F$-subnormal (submodular) in $G$ if there is a subgroup chain $H=H_0\le \ H_1 \le \ \ldots \le H_i \leq H_{i+1}\le \ldots \le \ H_n=G$ such that for every $i$ either $H_{i}$ is normal in $H_{i+1}$ or $H_{i+1}^\mathfrak{F} \le H_i$ ($H_i$ is a modular subgroup of $H_{i+1}$, respectively). We prove that a primary subgroup $R$ of a group $G$ is submodular in $G$ if and only if $R$ is $\mathrm{K}\mathfrak U_1$-subnormal in $G$. Here $\mathfrak{U}_1$ is the class of all supersolvable groups of square-free exponent. In addition, for a solvable subgroup-closed formation $\mathfrak{F}$, every solvable $\mathrm{K}\mathfrak{F}$-subnormal subgroup of a group $G$ is contained in the solvable radical of $G$.