Group in which every characteristic submonoid is a subgroup

Definition
A group $$G$$ is termed a group in which every characteristic submonoid is a subgroup if it satisfies the following equivalent conditions:


 * 1) Every characteristic subset that is a submonoid under the group multiplication is a subgroup, and hence a defining ingredient::characteristic subgroup, of the whole group.
 * 2) Every characteristic subset that is nonempty and is a subsemigroup under the group multiplication is a subgroup, and hence a characteristic subgroup, of the whole group.

Facts

 * Characteristic submonoid of group not implies subgroup shows that this property is not tautological.