ACIC implies nilpotent (finite groups)

Property-theoretic statement
For finite groups, the property of being ACIC is stronger than the property of being nilpotent.

Verbal statement
Any finite ACIC-group is nilpotent.

Automorph-conjugate subgroup
A subgroup $$H$$ of a group $$G$$ is termed an automorph-conjugate subgroup if for every automorphism $$\sigma$$ of $$G$$, $$H$$ and $$\sigma(H)$$ are conjugate subgroups.

ACIC-group
A group is termed ACIC if every automorph-conjugate subgroup is characteristic, or equivalently, any automorph-conjugate subgroup is normal.

Finite nilpotent group
A finite group is termed nilpotent if all Sylow subgroups are normal (this is just one of the formulations of nilpotence for finite groups). The definition breaks down for infinite groups).

Facts used

 * In a finite group, any Sylow subgroup is automorph-conjugate.
 * The definition of finite nilpotent group given above: a finite nilpotent group is a finite group in which every Sylow subgroup is normal.

Related facts

 * Frattini subgroup is nilpotent
 * Frattini subgroup is ACIC
 * Abelian implies ACIC

Proof using subgroup property collapse
For any finite group, we have:

Sylow $$\implies$$ automorph-conjugate

And for an ACIC-group, we have:

Automorph-conjugate $$\implies$$ normal

Thus, for a finite ACIC-group, we have:

Sylow $$\implies$$ Normal

Which is precisely the condition for being a finite nilpotent group.