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In this paper we prove that every nonabelian finite 2-group with a cyclic commutator subgroup has a noninner automorphism of order two fixing either Φ(G) or Z(G) elementwise. This, together with a result of Peter Schmid on regular p-groups, extends our result to the class of nonabelian finite p-groups with a cyclic commutator subgroup. 2010 Mathematics subject classification: primary 20D45; secondary 20D15.doi:10.1017/s0004972712000706 fatcat:jsnfisa42vgjdh4ezfzhvdllnq