Galois correspondence-closed subgroup property

Definition
A Galois correspondence-closed subgroup property is a subgroup property which arises in the following manner.

We start with a rule which, for every group, gives a binary relation between the group and another set constructed canonically from the group. The rule must be isomorphism-invariant, in the sense that any isomorphism of groups respects the binary relation.

The subgroup property we now get is the property of being a subgroup, which is also a closed subset of the group under the Galois correspondence induced by the binary relation.

Weaker metaproperties

 * Strongly intersection-closed subgroup property
 * Intersection-closed subgroup property
 * Identity-true subgroup property