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

Statement
Suppose $$G$$ is a finite group and $$H$$ is an fact about::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 fact about::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