Let $K$ be a group, $G \unlhd K$ be a finite normal subgroup of even order, and let $\langle h \rangle<K$ be an infinite cyclic subgroup, so that they fit into a short exact sequence $$0\to G\to K \to \langle h \rangle \to 0.$$

Question: when can $K$ modulo the normal subgroup generated by $\langle h \rangle$ be trivial?