Left-extensibility-stable subgroup property

Symbol-free definition
A subgroup property is termed left-extensibility-stable if it can be expressed in the function restriction formalism with the left side being extensibility-stable.

Definition with symbols
A subgroup property $$p$$ is termed left-extensibility-stable if we can write:

$$p = a \to b$$

where $$a$$ is an extensibility-stable function property.

The above symbols mean that:

and $$f$$ is a function on $$G$$ satisfying $$a$$, then there is a function $$f'$$ on $$K$$ satisfying $$a$$ such that the restriction of $$f'$$ to $$G$$ is $$f$$.
 * A subgroup $$H$$ has property $$p$$ in a group $$G$$ if and only if every function from $$G$$ to itself satisfying property $$a$$ restricts to a function from $$H$$ to itself satisfying property $$b$$.
 * Property $$a$$ being extensibility-stable means the following: whenever $$G \le K$$ are groups,

In terms of the left expressibility operator
The metaproperty of being a left-extensibility-stable subgroup property is obtained by applying the left expressibility operator to the function metaproperty of being extensibilility-stable.

Stronger metaproperties

 * Extensibility-stable invariance property
 * Left-inner subgroup property

Weaker metaproperties

 * Intermediate subgroup condition: