2-subnormal not implies hypernormalized

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

Stronger facts

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

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.