Cyclic normal implies potentially verbal in finite

Statement with symbols
Suppose $$K$$ is a finite group and $$H$$ is a cyclic normal subgroup of $$K$$. In other words, $$H$$ is a normal subgroup of $$K$$ that is also a cyclic group. Then, $$H$$ is a potentially verbal subgroup of $$G$$: there exists a group $$G$$ containing $$K$$ such that $$H$$ is a verbal subgroup of $$G$$. In fact, we can choose $$G$$ to itself be a finite group.

Related facts

 * Cyclic normal implies finite-pi-potentially verbal in finite
 * Homocyclic normal implies finite-pi-potentially fully invariant in finite
 * Central implies potentially verbal in finite
 * Homocyclic normal implies potentially fully invariant in finite
 * Central implies finite-pi-potentially verbal in finite
 * Central implies finite-pi-potentially characteristic in finite

Facts used

 * 1) uses::Cyclic normal implies finite-pi-potentially verbal in finite

Proof
The proof follows directly from fact (1), which is a somewhat stronger formulation.