Intermediate automorph-conjugacy is normalizer-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., intermediately automorph-conjugate subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)
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Statement

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. Automorph-conjugacy is normalizer-closed
2. Intermediately operator preserves normalizer-closedness

Proof

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