Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristic

Statement
Suppose $$G$$ is a group and $$H$$ is a nontrivial fact about::normal subgroup of $$G$$. Then, there exists a group $$K$$ containing $$G$$ such that $$H$$ is a fact about::characteristic subgroup of $$K$$ but not an intermediately characteristic subgroup of $$K$$ (i.e., it is not characteristic in every intermediate subgroup).

Facts used

 * 1) uses::Every nontrivial normal subgroup is potentially normal-and-not-characteristic
 * 2) uses::NPC theorem: This statems that every normal subgroup is potentially characteristic.

Proof
Given: A group $$G$$, a nontrivial normal subgroup $$H$$ of $$G$$.

To prove: There exists a group $$K$$ containing $$G$$ such that $$H$$ is characteristic in $$K$$ but not characteristic in every intermediate subgroup.

Proof:


 * 1) By fact (1), there exists a group $$L$$ containing $$G$$ such that $$H$$ is normal and not characteristic in $$L$$.
 * 2) By fact (2), there exists a group $$K$$ containing $$L$$ such that $$H$$ is characteristic in $$K$$.

Thus, $$H$$ is characteristic in $$K$$ but not in the intermediate subgroup $$L$$, completing the proof.