Central implies finite-pi-potentially verbal in finite

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Central subgroup (?)) must also satisfy the second subgroup property (i.e., Finite-pi-potentally verbal subgroup (?)). In other words, every central subgroup of finite group is a finite-pi-potentally verbal subgroup of finite group.
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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 Verbal subgroup (?) of $K$.

Related facts

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.