Normalizing join-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Definition

A subgroup property $p$ is termed normalizing join-closed if whenever $H,K\leq G$ 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 $p$ .