Normalizing join-closed subgroup property

Definition
A subgroup property $$p$$ is termed normalizing join-closed if whenever $$H, K \le G$$ are subgroups satisfying property $$p$$ such that $$K$$ is contained in the normalizer $$N_G(H)$$, the product of subgroups $$HK$$ (which is the same as the join of subgroups $$\langle H, K \rangle$$) also satisfies property $$p$$.

Stronger metaproperties

 * Weaker than::Join-closed subgroup property
 * Weaker than::Finite-join-closed subgroup property