Central implies potentially verbal in finite

Statement
Suppose $$G$$ is a finite group and $$H$$ is a central subgroup of $$G$$. Then, there exists a finite group $$K$$ containing $$G$$ such that $$H$$ is a fact about::verbal subgroup of $$K$$.

Stronger facts and other corollaries of those stronger facts

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

Weaker facts

 * Abelian direct factor implies potentially verbal in finite: The proof is essentially the same, using direct products instead of central products.

Other related facts

 * Cyclic normal implies finite-pi-potentially verbal in finite, cyclic normal implies potentially verbal in finite
 * Homocyclic normal implies finite-pi-potentially fully invariant in finite, homocyclic normal implies potentially fully invariant in finite

Facts used

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

Proof
The result follows directly from fact (1), which is a stronger version of the same statement.