Central implies potentially fully invariant in finite

Statement
Suppose $$G$$ is a finite group and $$H$$ is a central subgroup of $$G$$. In other words, $$H$$ is a subgroup of $$G$$ contained in the center of $$G$$. Then, there exists a group $$K$$ containing $$G$$ such that $$H$$ is a fully invariant subgroup of $$K$$.

Stronger facts

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

Other related facts

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