2-subnormal not implies hypernormalized

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) need not satisfy the second subgroup property (i.e., hypernormalized subgroup)
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Statement

Verbal statement

A 2-subnormal subgroup of a group need not be hypernormalized.

Related facts

Stronger facts

• Abnormal normalizer and 2-subnormal not implies normal: In fact, the same example used here works for that as well.

Proof

An example in the symmetric group on four letters

Let $G$ be the symmetric group on four letters $\{1,2,3,4\}$ and $H$ be the two-element subgroup generated by $(13)(24)$.

Then, $H$ is normal in the subgroup $K=\{(),(12)(34),(13)(24),(14)(23)\}$, which is normal in $G$. So $H$ is 2-subnormal in $G$.

On the other hand, the normalizer $N_G(H)$ is a dihedral subgroup of order eight, which is a self-normalizing subgroup.