Elementarily equivalently embedded subgroups

Definition
Let $$G$$ be a group and $$H_1$$ and $$H_2$$ be two subgroups. Consider the theory of the group $$G$$ along with a rule for membership in $$H_1$$, and correspondingly consider a theory for $$G$$ with a rule for membership in $$H_2$$. If these two theories are elementarily equivalent, then we say that $$H_1$$ and $$H_2$$ are elementarily equivalently embedded.

Stronger relations

 * Weaker than::Conjugate subgroups
 * Weaker than::Automorphic subgroups