Finite p-groups with few p-automorphisms

Let G be any non-cyclic group of order p(n), where n greater than or equal to 3. It is conjectured that in this case \G \ \ Aut(G)\ (in other words \G \ less than or equal to \ Aut(G)\p). In this paper we describe completely the p-groups G such that \ Aut(G)\ (p) = \G \ and such that G is either abe lian or of maximal class. In particular, we answer a question of Y. Berkovi ch.