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 $H\leq K\leq G$ 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.