Normal-extensible not implies inner

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-extensible automorphism) need not satisfy the second subgroup property (i.e., inner automorphism)
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Statement

We can have a group $G$ and a normal-extensible automorphism $\sigma$ of $G$ that is not an inner automorphism.

In fact, there exist groups for which it is not true that every automorphism is inner, but for which it is still true that every automorphism is normal-extensible.

Facts used

1. Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Proof

Given fact (1), we simply need to exhibit a group satisfying the condition of being centerless and maximal in its automorphism group. One example is the alternating group of degree four.