Strongly join-closed subgroup property

Definition
A subgroup property $$p$$ is termed strongly join-closed if for any (possibly empty, possibly finite and possibly infinite) collection $$H_i, i \in I$$ of subgroups of a group $$G$$ such that each $$H_i$$ satisfies $$p$$ in $$G$$, the join of subgroups $$\langle H_i \rangle_{i \in I}$$ also satisfies $$p$$ in $$G$$.

By convention, if $$I$$ is empty, the join is taken as the trivial subgroup. In particular, a subgroup property is strongly join-closed if and only if it is join-closed and trivially true (i.e., always satisfied by the trivial subgroup).