Normal-extensible not implies extensible

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., normal-extensible automorphism) need not satisfy the second automorphism property (i.e., extensible automorphism)
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Statement

There exists a group $G$ and a normal-extensible automorphism $\sigma$ of $G$ such that $\sigma$ is not an extensible automorphism.

Facts used

1. Finite-extensible implies class-preserving
2. Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Proof

Let $G$ be the alternating group:A4: in other words, $G$ is the alternating group on the set $\{1,2,3,4\}$. The automorphism group $K$ of $G$ can be identified naturally with the symmetric group on $\{1,2,3,4\}$, with $G$ embedded in it as inner automorphisms. By fact (2), every automorphism of $G$ (including the outer automorphisms) is normal-extensible.

On the other hand, none of the outer automorphisms of $G$ preserves conjugacy classes in $G$. Thus, by fact (1), none of the outer automorphisms of $G$ is finite-extensible. In other words, for any outer automorphism $\sigma$ of $G$, there exists a finite group $L$ containing $G$ such that $\sigma$ does not extend to $L$. In particular, $\sigma$ is not extensible, and we are done.