Normal-extensible automorphisms problem

Statement
An automorphism $$\sigma$$ of a group $$G$$ is termed a normal-extensible automorphism if, whenever $$G$$ is a normal subgroup of a group $$K$$, there is an automorphism $$\sigma'$$ of $$K$$ whose restriction to $$G$$ is $$\sigma$$.

The normal-extensible automorphisms of any group form a subgroup of the automorphism group. Further, this subgroup contains the group of inner automorphisms, since every inner automorphism is normal-extensible. In fact, it contains the group of extensible automorphisms as well.

The normal-extensible automorphisms problem is the problem of characterizing the group of normal-extensible automorphisms of a group. There are two extreme cases for normal-extensible automorphisms:


 * Group in which every automorphism is normal-extensible
 * Group in which every normal-extensible automorphism is inner

A group in which every automorphism is inner is at both extremes.

A somewhat intermediate case that is also important is group in which every normal-extensible automorphism is normal.

Groups in which every automorphism is normal-extensible
A group in which every automorphism is inner obviously satisfies the additional condition that every automorphism of the group is normal-extensible. However, there are examples of groups with outer automorphisms in which every automorphism is normal-extensible. Two basic 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

Here are some examples of this:


 * In dihedral group:D8, every automorphism is center-fixing and the inner automorphism group is maximal in the automorphism group. Thus, every automorphism is normal-extensible.
 * Any alternating group $$A_n$$, $$n \ne 1,2,6$$, satisfies the conditions of being centerless and maximal in its automorphism group. Thus, every automorphism of $$A_n$$ is normal-extensible.
 * Suppose $$S$$ is a simple non-abelian complete group. Then, $$S \times S$$ is of index two in its automorphism group.

These examples show that:


 * Normal-extensible not implies inner
 * Normal-extensible not implies normal: The examples of the dihedral group and the square of a simple non-abelian complete group show that there can exist normal-extensible automorphisms that are not normal automorphisms, i.e., they do not send normal subgroups to themselves.

The second fact can be restated as follows: a normal subgroup need not be a normal-extensible automorphism-invariant subgroup. This has the following corollaries:


 * Normal not implies normal-potentially characteristic: We can have a normal subgroup $$H$$ of a group $$G$$ such that there is no group $$K$$ containing $$G$$ as a normal subgroup and $$H$$ as a characteristic subgroup.
 * Normal not implies normal-potentially relatively characteristic