Intermediately automorph-conjugate implies intermediately characteristic in nilpotent

Statement
In a nilpotent group, any intermediately automorph-conjugate subgroup is intermediately characteristic.

Converse
The converse is clearly true in all groups: intermediately characteristic implies intermediately automorph-conjugate. In fact, in any group, characteristic implies automorph-conjugate.

Removing the intermediately operator
It is not in general true in a nilpotent group that every automorph-conjugate subgroup is characteristic. In fact, a group satisfying the property that every automorph-conjugate subgroup is characteristic is termed an ACIC-group. Some facts:


 * ACIC implies nilpotent (finite groups)
 * Nilpotent not implies ACIC

Facts used

 * 1) Intermediately automorph-conjugate implies intermediately normal-to-characteristic
 * 2) Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

Proof
The proof follows directly by combining facts (1) and (2).