Order-automorphic subgroup

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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 automorphic subgroups.