Single-input-expressible subgroup metaproperty

Definition
A subgroup metaproperty $$\alpha$$ is termed single-input-expressible if it admits the following description. Consider a procedure $$P$$ that takes as input a pair of a group and a subgroup, and outputs a collection of group-subgroup pairs.

A subgroup property $$p$$ satisfies $$\alpha$$ if and only if, whenever the starting pair satisfies property $$p$$, all the pairs in the resulting collection also satisfy property $$p$$.

Weaker metaproperties

 * Stronger than::Conjunction-closed subgroup metaproperty
 * Stronger than::Disjunction-closed subgroup metaproperty
 * Stronger than::Finite-input-expressible subgroup metaproperty