Double coset-separated subgroup

Symbol-free definition
A subgroup of a group is termed double coset-separated if whenever two left cosets of the subgroup are conjugate (that is, there is an inner automorphism of the group mapping one bijectively to the other) then the two left cosets must lie in the same double coset of the subgroup.

Stronger properties

 * Self-normalizing subgroup
 * Abelian-quotient subgroup: In fact, this is the conjunction of being normal and double coset-separated
 * Weakly cocentral subgroup
 * Subgroup of double coset index two

Metaproperties
From the definition, it is clear that if $$H$$ is double coset-separated in $$G$$, and $$K$$ is any intermediate subgroup, then $$H$$ is also double coset-separated in $$K$$.