Left-inner implies intermediate subgroup condition

This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., left-inner subgroup property) must also satisfy the second subgroup metaproperty (i.e., intermediate subgroup condition)
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Statement

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

Definitions used

Left-inner subgroup property

Further information: 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

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

Proof

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