Dedekind implies ACIC

Property-theoretic statement
The group property of being a Dedekind group is stronger than the group property of being an ACIC-group.

Dedekind group
A group is said to be Dedekind if every subgroup is normal.

ACIC-group
A group is said to be ACIC if every automorph-conjugate subgroup is normal, or equivalently, every automorph-conjugate subgroup is characteristic.

Proof using subgroup property collapse
In a Dedekind group, every subgroup is normal. In particular, every automorph-conjugate subgroup is normal, and hence, the group is an ACIC-group.