Abnormal normalizer and 2-subnormal not implies normal

Statement
A fact about::2-subnormal subgroup with abnormal normalizer, i.e., a 2-subnormal subgroup whose normalizer in the whole group is an abnormal subgroup, need not be a normal subgroup.

Corollaries

 * 2-subnormal not implies hypernormalized
 * Abnormal normalizer not implies pronormal

Other related facts

 * Normalizer of 2-subnormal subgroup may have arbitrarily large subnormal depth

An example of the symmetric group on four letters
Let $$G = S_4$$ by the symmetric group on the set $$\{ 1,2,3,4\}$$, and $$H$$ be the two-element subgroup generated by the double transposition $$(1,2)(3,4)$$. Then, the normalizer $$N_G(H)$$ is the dihedral group of order eight, which is a maximal non-normal subgroup, and hence abnormal. Hence, $$H$$ has abnormal normalizer.

On the other hand, $$H$$ is contained in the subgroup $$K$$ of order four, comprising the identity and the double transpositions. $$H$$ is normal in $$K$$, and $$K$$ is normal in $$G$$, so $$H$$ is 2-subnormal in $$G$$.