Normal not implies normal-potentially characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., semi-strongly potentially characteristic subgroup)
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Statement

Verbal statement

It is possible to have a normal subgroup of a group that is not a semi-strongly potentially characteristic subgroup.

Statement with symbols

We can have a group $K$ with a subgroup $H$ such that $H$ is normal in $K$, but whenever $G$ is a group containing $K$ as a normal subgroup, $H$ is not a characteristic subgroup in $G$.

Facts used

1. Normal-extensible not implies normal

Proof

By fact (1), we can find a group $K$, a normal subgroup $H$ of $K$, and a normal-extensible automorphism $\sigma$ of $K$ such that $\sigma (H)\ne H$. Thus, for any group $G$ containing $K$ as a normal subgroup, $\sigma$ extends to an automorphism ${\sigma }^{\prime }$ of $G$. But then, ${\sigma }^{\prime }(H)=\sigma (H)\ne H$, so $H$ is not characteristic in $G$.