Homocyclic normal implies potentially fully invariant in finite

Statement
Suppose $$G$$ is a finite group and $$H$$ is a homocyclic normal subgroup of $$G$$: in other words, $$H$$ is a normal subgroup of $$G$$ as well as a homocyclic group: it is a direct product of isomorphic cyclic groups. Then, there exists a finite group $$K$$ containing $$G$$ such that $$H$$ is a fact about::fully invariant subgroup of $$K$$.

Facts used

 * 1) uses::Homocyclic normal implies finite-pi-potentially fully invariant in finite

Proof
The result follows directly from fact (1), which is a stronger version.