Finite groups with semi-subnormal Schmidt subgroups

A Schmidt group is a non-nilpotent group in which every proper subgroup is nilpotent. A subgroup $A$ of a group $G$ is semi-normal in $G$ if there exists a subgroup $B$ of $G$ such that $G=AB$ and $AB_1$ is a proper subgroup of $G$ for every proper subgroup $B_1$ of $B$. If $A$ is either subnormal in $G$ or is semi-normal in $G$, then $A$ is called a semi-subnormal subgroup of $G$. In this paper, we establish that a group $G$ with semi-subnormal Schmidt $\{2,3\}$-subgroups is $3$-soluble. Moreover, if all 5-closed Schmidt $\{2,5\}$-subgroups are semi-subnormal in $G$, then $G$ is soluble. We prove that a group with semi-subnormal Schmidt subgroups is metanilpotent.