Intermediate subgroup condition

Definition
A subgroup property $$p$$ is said to satisfy the intermediate subgroup condition if whenever $$H \le K \le G$$ are groups and $$H$$ satisfies $$p$$ in $$G$$, $$H$$ also satisfies $$p$$ in $$K$$.

Formalisms
Consider a procedure $$P$$ that takes as input a group-subgroup pair $$H \le G$$ and outputs all group-subgroup pairs $$H \le K$$ where $$K$$ is an intermediate subgroup of $$G$$ containing $$H$$. Then, the intermediate subgroup condition is the single-input-expressible subgroup property corresponding to procedure $$P$$. In other words, a subgroup property $$p$$ satisfies the intermediate subgroup condition if whenever $$H \le G$$ satisfies property $$p$$, all the pairs obtained by applying procedure $$P$$ to $$H \le G$$ also satisfy property $$p$$.

In terms of the intermediately operator
A subgroup property satisfies intermediate subgroup condition if and only if it is a fixed-point of the idempotent subgroup property modifier called the intermediately operator.

In terms of the potentially operator
A subgroup property satisfies intermediate subgroup condition if and only if it is a fixed-point of the idempotent subgroup property modifier called the potentially operator.

Conjunction implications

 * Any left-realized subgroup property satisfying intermediate subgroup condition must be identity-true.