Potentially characteristic 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., potentially characteristic subgroup) need not satisfy the second subgroup property (i.e., normal-potentially characteristic subgroup)
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Statement

It is possible to have a potentially characteristic subgroup that is not a normal-potentially characteristic subgroup.

Related facts

Weaker facts

• Normal not implies normal-potentially characteristic
• Normal not implies characteristic-potentially characteristic
• Potentially characteristic not implies characteristic-potentially characteristic

Facts used

1. Potentially characteristic not implies normal-extensible automorphism-invariant, which in turn follows from Normal not implies normal-extensible automorphism-invariant in finite and finite normal implies potentially characteristic.
2. Normal-potentially characteristic implies normal-extensible automorphism-invariant

Proof

The proof follows 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 potentially characteristic in $G$: for instance, we can realize $G$ as the $2$-Sylow subgroup of the symmetric group of degree four, in such a way that $H$ becomes characteristic.