On noninner 2-automorphisms of finite 2-groups. (Q2922934)

The authors consider finite \(2\)-groups \(G\) and they are interested in seeing if \(G\) admits noninner \(2\)-automorphisms of minimal possible order fixing the Frattini subgroup elementwise. They succeed to prove that if \(G\) has coclass 2, or if \((G,Z(G))\) is a Camina pair, then \(G\) has a noninner automorphism of order 2 or 4 acting trivially on the Frattini subgroup of \(G\).NEWLINENEWLINE An example is given, which produces a group \(G\) of order 32, of coclass 2, such that \((G,Z(G))\) is a Camina pair and which has no noninner automorphism of order 2 centralizing the Frattini subgroup of \(G\). This explains the appearence of the order 4 noninner automorphisms and shows that the authors' statements are in some sense the best possible.