Normal-extensible not implies extensible

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) uses::Finite-extensible implies class-preserving
 * 2) uses::Centerless and maximal in automorphism group implies every automorphism is normal-extensible

Proof
Let $$G$$ be the particular example::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.