Finite-relative-intersection-closed subgroup property

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

Suppose $$H,K$$ are subgroups of a group $$G$$ such that $$H$$ satisfies property $$p$$ in $$G$$ and $$K$$ satisfies property $$p$$ in some subgroup of $$G$$ containing both $$H$$ and $$K$$. Then, $$H \cap K$$ satisfies property $$p$$ in $$G$$.

Stronger metaproperties

 * Weaker than::Relative-intersection-closed subgroup property

Weaker metaproperties

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

Facts

 * A transitive subgroup property that also satisfies the transfer condition is finite-relative-intersection-closed.