Potential-tautological subgroup property

Symbol-free definition
A subgroup property is said to be potential-tautological if its image under the potentially operator is the tautology subgroup property viz the property of being any subgroup.

Definition with symbols
A subgroup property $$p$$ is said to be potential-tautological if whenever $$H \le G$$ is a subgroup, we can find a group $$K$$ containing $$G$$ such that $$H$$ satisfies property $$p$$ in $$K$$.

Weaker metaproperties

 * Potential-nontrivial-tautological subgroup property: Every nontrivial subgroup has property potentially $$p$$.
 * Potential-proper-tautological subgroup property: Every proper subgroup has property potentially $$p$$.