Transitively normal subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, formula_1 is a transitively normal subgroup of formula_2 if for every formula_3 normal in formula_1, we have that formula_3 is normal in formula_2.
An alternate way to characterize these subgroups is: every "normal subgroup preserving automorphism" of the whole group must restrict to a "normal subgroup preserving automorphism" of the subgroup.
Here are some facts about transitively normal subgroups: