Left-inner implies intermediate subgroup condition

Statement
Any left-inner subgroup property satisfies the intermediate subgroup condition.

Left-inner subgroup property
A subgroup property $$p$$ is termed left-inner if there exists a property $$\alpha$$ of functions from a group to itself such that $$p$$ can be written using the function restriction expression:

inner automorphism $$\to$$ $$\alpha$$

In other words, a subgroup $$H$$ of a group $$G$$ satisfies property $$p$$ in $$G$$ if and only if every inner automorphism of $$G$$ restricts to a function from $$H$$ to itself that satisfies $$\alpha$$.

Intermediate subgroup condition
A subgroup property $$p$$ is said to satisfy the intermediate subgroup condition if, for any groups $$H \le K \le G$$ such that $$H$$ satisfies $$p$$ in $$G$$, $$H$$ also satisfies $$p$$ in $$K$$.

Facts used

 * 1) uses::Inner is extensibility-stable: An inner automorphism of a subgroup can be extended to an inner automorphism of the whole group.
 * 2) uses::Left-extensibility-stable implies intermediate subgroup condition

Proof
The proof follows by combining facts (1) and (2).