The authors continue their investigation of $S$-groups, i.e. the groups $G$, in which every subgroup $H$ is $f$-subnormal (there exists an $f$-series from $H$ to $G$, that is a finite series $H=H_0\le H_1\le\cdots\le H_n=G$ such that $|H_i:H_{i-1}|$ is finite or $H_{i-1}$ is normal in $H_i$).\par The main results of the article are the following theorems: Theorem 1. A torsion $S$-groups is an extension of a finite group by a group in which all subgroups are subnormal. Theorem 2. The primary components of the subgroup generated by the nilpotent residuals of the finitely generated subgroups of an $S$-group $G$ are finite. [Igor Subbotin (Los Angeles)]