Monadic second-order characteristic subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a monadic second-order characteristic subgroup if there is no other subgroup $$K$$ of $$G$$ such that the monadic second-order theories of the group-subgroup pairs $$(G,H)$$ and $$(G,K)$$ coincide. In other words, $$H$$ can be distinguished from any other subgroup of $$G$$ using monadic second-order logic in the pure theory of the group $$G$$.