Extensibility-stable function property
A property 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 in the subgroup, there is a function on the group satisfying property that extends the given function.
Definition with symbols
A property of functions from a group to itself is said to be extensibility-stable if, given any groups , and a function satisfying property in , there is a function such that satisfies and such that the restriction of to is .
In terms of the extensibility operator
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. For full proof, refer: Inner is extensibility-stable