Group having an abelian conjugate-dense subgroup

Definition
A group is termed a group having an abelian conjugate-dense subgroup if there exists a subgroup $$H$$ of $$G$$ such that $$H$$ is abelian as a group and is conjugate-dense in $$G$$, i.e., every element of $$G$$ is conjugate to an element of $$H$$.

Note that for a finite group, this is equivalent to being an abelian group (hence a finite abelian group) because union of all conjugates is proper and hence there is no proper conjugate-dense subgroup.