Join-transiter

Symbol-free definition
Let $$p$$ be a subgroup property. Then the join-transiter of $$p$$ is the property of being a subgroup such that its join with any subgroup having property $$p$$ again has property $$p$$ in the whole group.

Definition with symbols
Let $$p$$ be a subgroup property. Then the join-transiter of $$p$$ is the property $$q$$ such that: A subgroup $$H$$ has property $$q$$ in $$G$$ if and only if whenever $$K \le G$$ has property $$p$$, $$$$ has property $$p$$ in $$G$$.