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 $G$.

Related facts

Stronger facts

• Normal not implies semi-strongly potentially relatively characteristic
• Potentially characteristic not implies semi-strongly potentially characteristic
• Potentially characteristic not implies semi-strongly potentially relatively characteristic

Weaker facts

• Normal not implies strongly potentially characteristic

Facts used

1. Normal not implies normal-extensible automorphism-invariant
2. Semi-strongly potentially characteristic implies normal-extensible automorphism-invariant

Proof

The proof follows directly from facts (1) and (2).