Extensibility-stable function property

Symbol-free definition
A property $$p$$ of functions from a group to itself is said to be extensibility-stable if, given any subgroup of a group, and a function on the subgroup satisfying property $$p$$ in the subgroup, there is a function on the group satisfying property $$p$$ that extends the given function.

Definition with symbols
A property $$p$$ of functions from a group to itself is said to be extensibility-stable if, given any groups $$H \le G$$, and a function $$f: H \to H$$ satisfying property $$p$$ in $$H$$, there is a function $$f':G \to G$$ such that $$f'$$ satisfies $$p$$ and such that the restriction of $$f'$$ to $$H$$ is $$f$$.

In terms of the extensibility operator
Extensibility-stable function properties are precisely those function properties that are fixedp oints underthe extensibility operator on the function property space.

Inner automorphism
Any inner automorphism of a subgroup lifts to an inner automorphism of the whole group. This is because we can take the same conjugating element from the subgroup and use it to define a conjugation on the whole group. Note that since for a given inner automorphism, the choice of conjugating element is not unique, the lift is in general not unique.