Intermediately subnormal-to-normal is normalizer-closed

Statement
Suppose $$H$$ is an intermediately subnormal-to-normal subgroup of a group $$G$$. Then, the normalizer $$N_G(H)$$ is also an intermediately subnormal-to-normal subgroup of $$G$$.

Facts used

 * 1) Subnormal-to-normal is normalizer-closed
 * 2) Intermediately operator preserves normalizer-closedness

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