Torsion-free groups with factor-groups on their hypercenter which are periodic

Assume that $G$ is a torsion-free group, $Z_k(G)$ is the $k$-th term of the upper central series of $G$, and $\overline{G}_k=G/Z_k(G)$ is a nontrivial periodic group. Then every finite subgroup of $\overline{G}_k$ is nilpotent of class not higher than $k$; the group $k\geqslant2$ contains an infinite subgroup with $k$ generators if $\overline{G}_k$ and two generators if $k=1$. Moreover any nontrivial invariant subgroup of $\overline{G}_k$ is infinite. All elements of $\overline{G}_k$ are of odd order. This assertion is generalized.