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., normal-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 normal-potentially characteristic subgroup.

Statement with symbols

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

Related facts

Stronger facts

• Normal not implies normal-potentially relatively characteristic

Weaker facts

• Normal not implies characteristic-potentially characteristic

Facts used

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

Proof

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

Example of the dihedral group

Further information: dihedral group:D8

Let $G$ be the dihedral group of order eight, and $H$ be one of the Klein four-subgroups.

• $H$ is not a normal-potentially characteristic subgroup of $G$: Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of $G$ can be extended to an automorphism of $K$ for any group $K$ containing $G$ as a normal subgroup. But since there is an automorphism of $G$ not sending $H$ to itself, $H$ cannot be characteristic in $K$.
• $H$ is normal in $G$: This is obvious.