Group having an abelian contranormal subgroup

Definition
A group $$G$$ is termed a group having an abelian contranormal subgroup if there exists a subgroup $$H$$ of $$G$$ that is abelian as a group and is a defining ingredient::contranormal subgroup of $$G$$, i.e., the defining ingredient::normal closure of $$H$$ in $$G$$ equals $$G$$.