Left-extensibility-stable implies intermediate subgroup condition

Statement
Let $$p$$ be a left-extensibility-stable subgroup property, viz a subgroup property that can be written as $$a \to b$$ where $$a$$ is an extensibility-stable function property.

Then, $$p$$ satisfies the intermediate subgroup condition, or equivalently, whenever $$H \le K \le G$$ with $$H$$ satisfying $$p$$ in the whole of $$G$$, $$H$$ also satisfies $$p$$ in $$K$$.

Examples
Some examples of subgroup properties that are left-extensibility-stable and, on account of this, satisfy the intermediate subgroup condition, are:


 * Normal subgroup
 * Central factor

Proof
Let $$p$$ be a left-extensibility-stable subgroup property and $$H \le K \le G$$ be groups such that $$H$$ satisfies $$p$$ in $$G$$. We need to show that $$H$$ satisfies $$p$$ in $$K$$.

Let $$\sigma$$ be a function on $$K$$ satisfying property $$p$$ in $$K$$. Then, we need to show that $$\sigma$$ restricts to a function on $$H$$ which satisfies $$b$$ in $$H$$.

Since $$a$$ is an extensibility-stable function property, there exists a function $$\sigma'$$ on $$G$$ whose restriction to $$K$$ is $$\sigma$$. Now, since $$H$$ satisfies property $$a \to b$$ in $$G$$, the restriction of $$\sigma'$$ to $$H$$ is well-defined and satisfies property $$b$$ in $$H$$.

But the restriction to $$H$$ of $$\sigma'$$ is the same as the restriction to $$H$$ of $$\sigma$$. Hence, we have shown that the restriction to $$H$$ of $$\sigma$$ is well-defined and satisfies property $$b$$ over $$H$$.