Fully normalized subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be fully normalized if every automorphism of the subgroup lifts to an inner automorphism of the whole group. Another way of putting this is that the natural embedding from the Weyl group of the subgroup to its automorphism group is surjective.
In symbols, a subgroup formula_1 is fully normalized in formula_2 if, given an automorphism formula_3 of formula_1, there is a formula_5 such that the map formula_6, when restricted to formula_1 is equal to formula_3.
Some facts: