Central implies potentially verbal in finite

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

Related facts

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. 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.