Automorph-commensurable subgroup

Symbol-free definition
A subgroup of a group is termed an automorph-commensurable subgroup if it is commensurable with all its defining ingredient::automorphic subgroups.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed an automorph-commensurable subgroup if, for any automorphism $$\sigma$$ of $$G$$, $$H \cap \sigma(H)$$ has finite index in both $$H$$ and $$\sigma(H)$$.