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We consider the class N of groups in which the normaliser of every subgroup is normal, and the class C of groups in which the commutator subgroup normalises every subgroup. It is clear that C ⊆ 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 ≥ 5, and we give an example showing that this is not true for p = 2.