Potentially characteristic not implies normal-extensible automorphism-invariant

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-extensible automorphism-invariant subgroup)
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Statement

A potentially characteristic subgroup of a group need not be a normal-extensible automorphism-invariant subgroup.

Facts used

1. Normal not implies normal-extensible automorphism-invariant in finite: Actually, we need a slightly stronger version of this statement. Namely, we need that there is a normal-extensible automorphism of a finite group that is not a normal automorphism. The example given in the proof is a finite group, so the proof actually shows the stronger version.
2. Finite normal implies potentially characteristic, or, alternatively, Finite NPC theorem

Proof

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