Normal-extensible not implies normal

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., normal automorphism)
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Statement

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
• Normal-extensible not implies inner
• Normal-extensible not implies extensible

Applications

• Normal not implies semistrongly potentially characteristic
• Normal not implies strongly potentially characteristic

Facts used

1. Centerless and maximal in automorphism group implies every automorphism is normal-extensible
2. Automorphism group of direct power of simple non-abelian group is wreath product of automorphism group and symmetric group: Let $S$ be a simple non-abelian group and $S^n$ be a direct product of $S$ with itself $n$ times. The automorphism group of $S^n$ is $\mathrm{A}\mathrm{u}\mathrm{t}(S)\wr \mathrm{S}\mathrm{y}\mathrm{m}(n)$, where the latter has the usual action on $n$ letters.

Proof

Further information: general linear group:GL(3,2)

Let $S=GL(3,2)$, i.e., $S$ is the general linear group of order three over the field of two elements. Consider $G=S\times S$.

We know that $S$ is simple and non-abelian (hence centerless) and every automorphism of $S$ is inner, so $S=\mathrm{A}\mathrm{u}\mathrm{t}(S)$.

By fact (2), the automorphism group of $G$ is $\mathrm{A}\mathrm{u}\mathrm{t}(S)\wr \mathbb{Z}/2\mathbb{Z}$. Because every automorphism of $S$ is inner, this is $G⋊\mathbb{Z}/2\mathbb{Z}$, with the action being the coordinate exchange automorphism. Thus, $G$ is centerless and maximal in its automorphism, so by fact (1), every automorphism of $G$ is normal-extensible.

Let $N$ be the normal subgroup of $G$ defined by the first direct factor, i.e., $N=S\times 1$. Let $\sigma$ be the coordinate exchange automorphism of $G$, i.e., $\sigma (a,b)=(b,a)$. Then, since every automorphism of $G$ is normal-extensible, $\sigma$ is normal-extensible. However, $\sigma (N)\ne N$, since $\sigma (N)=1\times S$ is the other direct factor.