Coset-conjugacy class

Definition with symbols
Let $$H$$ be a subgroup of $$G$$. A coset-conjugacy class of $$H$$ is the set of all left cosets of $$H$$ that are conjugate to a given left coset $$xH$$. Note here that we call two left cosets conjugate if there is an inner automorphism that maps one coset bijectively onto another.

Relation with double cosets
Any two left cosets of $$H$$ that lie in the same double coset of $$H$$ lie in the same coset-conjugacy class. In fact, two left cosets lie in the same double coset if and only if there is an inner automorphism arising from an element of $$H$$ that takes one to the other.

A subgroup for which two left cosets are conjugate if and only if they lie in the same double coset is termed a double coset-separated subgroup, Note that any self-normalizing subgroup is double coset-separated, and any normal double coset-separated subgroup must be an Abelian-quotient subgroup.