Finite p-groups with normal normalisers

We consider the class N of groups in which the normaliser of every subgroup is normal, and the class C in which the commutator subgroup normalises every subgroup. It is clear that C is contained in N, and it is known that groups in the class N are nilpotent of class at most 3. We show that every finite p-group in N is also in C, provided that p is greater than or equal to 5, and we give an example showing that this is not true for p=2.