Rough approximation of subgroup and conjugacy relation

Abstract

Let G be a group and H be a rough subgroup of G with respect to the conjugacy relation, which is considered as an equivalence relation. An internal edge of H is defined as the difference between H and its lower approximation. Let E be a non-empty subset of G. In this paper, we aim to answer the following questions: Can E represent an internal edge of some subgroup of G ( in other words, what are the conditions that E must satisfy in order to be an internal edge of some subgroup of G)? If the answer to this question is
yes, what is this subgroup, and is it unique or not?