Intermediate automorph-conjugacy is normalizer-closed

Property-theoretic statement
The property of being an intermediately automorph-conjugate subgroup is a normalizer-closed subgroup property: it is closed upon taking normalizers in the whole group.

Verbal statement
If $$H$$ is an intermediately automorph-conjugate subgroup of a group $$G$$ (i.e., $$H$$ is an automorph-conjugate subgroup in every intermediate subgroup of $$G$$), then $$N_G(H)$$ is also an intermediately automorph-conjugate subgroup of $$G$$.

Facts used

 * 1) uses::Automorph-conjugacy is normalizer-closed
 * 2) uses::Intermediately operator preserves normalizer-closedness

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