Transitivity-forcing operator

Definition
The transitivity-forcing operator is an operator that takes as input a subgroup property and outputs a group property. Applied to a subgroup property $$p$$ it returns a group property $$q$$ as follows:

$$G$$ satisfies property $$q$$ if and only if whenever $$H \le K \le G$$ and $$H$$ satisfies $$p$$ in $$K$$ and $$K$$ satisfies $$p$$ in $$G$$, then $$H$$ satisfies $$p$$ in $$G$$.

The word transitivity-forcing is because applied to a transitive subgroup property, it gives the group property which is true for all groups.