Potentially relatively characteristic equals normal

Statement

A subgroup of a group is termed potentially relatively characteristic if there is an embedding of the whole group in some bigger group such that every automorphism of the supergroup that restricts to an automorphism of the group, in fact restricts to an automorphism of the subgroup as well.

Then, a subgroup is normal if and only if it is potentially relatively characteristic.

Definition with symbols

A subgroup $H$ of a group $G$ is termed potentially relatively characteristic in $G$ if there is a group $K$ such that every automorphism of $K$ that restricts to an automorphism of $G$, also restricts to an automorphism of $H$. $H$ is normal in $G$ if and only if $H$ is potentially relatively characteristic in $G$.

Proof

Proof outline

The direction of potentially relatively characteristic implies normal is straightfoward.

For the other direction, the proof idea is as follows. Make the group $G$ act on the coset space of the normal subgroup $H$, and use this to get a homomorphism from the group to a symmetric group. A little trick can be used to get an injective homomorphism, and further, to ensure that the symmetric group is a complete group. Then, we show that inside this symmetric group, any automorphism that restricts to an automorphism of $G$ must also restrict to an automorphism of $H$.