Subgroup of finite abelian group implies finite-abelian-pi-potentially verbal

Statement
Suppose $$G$$ is a finite abelian group and $$H$$ is a subgroup of $$G$$. Then, there exists a finite abelian group $$K$$ containing $$G$$ and having no prime factors other than those of $$G$$, such that $$H$$ is a verbal subgroup of $$K$$.

In particular, every fact about::subgroup of finite abelian group is an fact about::abelian-potentially verbal subgroup, fact about::abelian-potentially fully invariant subgroup, and fact about::abelian-potentially characteristic subgroup.

Similar 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

Opposite facts

 * Subgroup of abelian group not implies abelian-potentially characteristic, subgroup of abelian group not implies abelian-extensible automorphism-invariant