Automorph-permutable subgroup

Symbol-free definition
A subgroup of a group is termed automorph-permutable if it permutes with any subgroup obtained as the image of it under an automorphism of the whole group.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed automorph-permutable if for any automorphism $$\sigma$$ of $$G$$, $$H\sigma(H) = \sigma(H)H$$.

Stronger properties

 * Permutable subgroup
 * Normal subgroup

Weaker properties

 * Conjugate-permutable subgroup