Abelian pronormal subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed an abelian pronormal subgroup of $$G$$ if it satisfies the following two conditions:


 * 1) $$H$$ is an abelian group.
 * 2) $$H$$ is a pronormal subgroup of $$G$$: If $$K$$ is a conjugate of $$H$$ in $$G$$, then $$H, K$$ are also conjugate in $$\langle H, K \rangle$$.