Relative-intersection-closed subgroup property

Definition
A subgroup property $$p$$ is termed relative-intersection-closed if it satisfies the following:

Suppose $$(I, <)$$ is a well-ordered set. Suppose $$H_i, i \in I$$ is a collection of subgroups of a group $$G$$, such that for any $$i \in I$$, $$H_i$$ satisfies property $$p$$ in some subgroup of $$G$$ containing both $$H_i$$ and $$\bigcap_{j < i} H_j$$. Then:

$$\bigcap_{i \in I} H_i$$

satisfies property $$p$$ in $$G$$.

Weaker metaproperties

 * Stronger than::Finite-relative-intersection-closed subgroup property
 * Stronger than::Transitive subgroup property
 * Stronger than::Intersection-closed subgroup property
 * Stronger than::Finite-intersection-closed subgroup property