Abelian hereditarily normal implies finite-pi-potentially verbal in finite

Statement

Suppose $G$ is a finite group and $H$ is an Abelian hereditarily normal subgroup (?) of $G$. In other words, $H$ is an abelian normal subgroup of $G$ such that the induced action of the quotient group $G/H$ on $H$ is by power automorphisms.

Then, $H$ is a Finite-pi-potentially verbal subgroup (?) of $G$. In other words, there exists a finite group $K$ containing $G$ such that all prime factors of the order of $K$ also divide the order of $G$, and such that $H$ is a verbal subgroup of $K$.

Related facts

• Central implies finite-pi-potentially verbal in finite
• Cyclic normal implies finite-pi-potentially verbal in finite
• Homocyclic normal implies finite-pi-potentially fully invariant in finite