Order-automorphic subgroup

Definition
Suppose $$G$$ is a group and $$H$$ is a finite subgroup of $$G$$. We say that $$H$$ is an order-automorphic subgroup of $$G$$ if every subgroup $$K$$ of $$G$$ having the same order as $$H$$ is related to $$H$$ via some automorphism, i.e., $$H$$ and $$K$$ are defining ingredient::automorphic subgroups.