Normal-extensible not implies normal

In terms of automorphism properties
A normal-extensible automorphism of a group (i.e., an automorphism that can always be extended for any embedding of the group as a normal subgroup of a bigger group) need not be a normal automorphism, i.e., it need not send every normal subgroup to itself.

In terms of subgroup properties
A normal subgroup of a group need not be a normal-extensible automorphism-invariant subgroup: i.e., there may be normal-extensible automorphisms of the group that do not leave the normal subgroup invariant.

Statement with symbols
We can have a group $$G$$ and a normal-extensible automorphism $$\sigma$$ of $$G$$ that is not a normal automorphism: in other words, there exists a normal subgroup $$N$$ of $$G$$ such that $$\sigma(N) \ne N$$.

Related facts

 * Centerless and maximal in automorphism group implies every automorphism is normal-extensible
 * Every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible
 * Normal-extensible not implies inner
 * Normal-extensible not implies extensible

Stronger facts

 * Normal not implies normal-extensible automorphism-invariant in finite: This is the stronger version, and the example outlined here in fact shows the stronger version.

Applications

 * Normal not implies semi-strongly potentially characteristic
 * Normal 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.