Normal not implies normal-extensible automorphism-invariant in finite

Statement with symbols
It is possible to have a finite group $$G$$, a normal subgroup $$N$$ of $$G$$, and a normal-extensible automorphism $$\sigma$$ of $$G$$ such that $$\sigma(N) \ne N$$.

Weaker facts

 * Normal-extensible not implies normal
 * Normal-extensible not implies extensible
 * Normal-extensible not implies inner

Applications

 * Normal not implies semi-strongly potentially relatively characteristic
 * Potentially characteristic not implies semi-strongly potentially relatively characteristic
 * Normal not implies semi-strongly potentially characteristic, normal not implies strongly potentially characteristic
 * Potentially characteristic not implies semi-strongly potentially characteristic, potentially characteristic not implies strongly potentially characteristic

Facts used

 * 1) uses::Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
 * 2) uses::Automorphism group of direct power of simple non-abelian group equals wreath product of automorphism group and symmetric group

Example of the dihedral group
Let $$G$$ be the dihedral group of order eight. Then, every automorphism of $$G$$ fixes every element of the center of $$G$$, and also, the inner automorphism group of $$G$$ is maximal in the automorphism group of $$G$$. Thus, by fact (1), every automorphism of $$G$$ is normal-extensible.

However, there is an automorphism of $$G$$ that interchanges the two normal Klein four-subgroups. Thus, these two normal subgroups are not invariant under this automorphism, and hence, we have an automorphism of $$G$$ that is normal-extensible but not normal.

Equivalently, the Klein four-subgroups are examples of normal subgroups that are not normal-extensible automorphism-invariant.

Example involving a simple complete group
Let $$S$$ be a simple complete group. In other words, $$S$$ is a centerless simple group such that every automorphism of $$S$$ is inner. Let $$G = S \times S$$. By fact (2), the automorphism group of $$G$$ is the wreath product of $$S$$ with the symmetric group of degree two, which has $$G$$, the inner automorphism group, as a subgroup of index two. Moreover, $$G$$ is centerless. Thus, by fact (1), we get that every automorphism of $$G$$ is normal-extensible.

However, the coordinate exchange automorphism of $$G$$, that interchanges the two copies of $$S$$, is not a normal automorphism because it interchanges these two normal subgroups. Thus, we have an example of a normal-extensible automorphism that is not normal.

Equivalently, either of the direct factors is an example of a normal subgroup that is not normal-extensible automorphism-invariant.