Central implies finite-pi-potentially verbal in finite

Statement
Suppose $$G$$ is a finite group and $$H$$ is a central subgroup of $$G$$, i.e., $$H$$ is contained inside the center of $$G$$. Then, there exists a finite group $$K$$ containing $$G$$ such that every prime divisor of the order of $$K$$ also divides the order of $$G$$, and $$H$$ is a fact about::verbal subgroup of $$K$$.

Weaker facts

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

Other related facts

 * Cyclic normal implies finite-pi-potentially verbal in finite
 * Homocyclic normal implies finite-pi-potentially fully invariant in finite
 * Normal not implies finite-pi-potentially characteristic in finite
 * Normal not implies potentially verbal
 * Normal not implies potentially fully invariant
 * Central and additive group of a commutative unital ring implies potentially iterated commutator subgroup in solvable group
 * NPC theorem: normal equals potentially characteristic
 * Finite NPC theorem: normal equals potentially characteristic in finite groups.