Conjugate-intersection-closed subgroup property

Symbol-free definition
A subgroup property is termed conjugate-intersection-closed if whenever a subgroup has the property, then any intersection of a family of conjugate subgroups of that subgroup also has that property.

Definition with symbols
We say that property $$p$$ is conjugate-intersection-closed if the following holds.

Suppose $$G$$ is a group, $$H$$ is a subgroup, and $$S$$ is any subset of $$G$$. Then, if $$H$$ has property $$p$$ in $$G$$ so does the group:

$$\bigcap_{g \in S} gHg^{-1}$$

Stronger metaproperties

 * Intersection-closed subgroup property
 * Automorph-intersection-closed subgroup property

Weaker metaproperties

 * Normal core-closed subgroup property