Finite \(p\)-groups with few \(p\)-automorphisms (Q2769823)

Let \(G\) be a noncyclic group of order \(p^n\geq p^3\). It is conjectured that in this case, \(|\Aut(G)|_p\geq|G|\). This conjecture was proved in many special cases. Using Fitting's formula for the order of the automorphism group of an Abelian \(p\)-group \(G\), the author shows that \(|\Aut(G)|_p\geq p^2|G|\) for \(n>3\), unless \(G\) is of type \((p^{n-1},p)\) (then \(|\Aut(G)|_p=|G|\)) or \(G\) is of type \((p^2,p^2)\) (then \(|\Aut(G)|_p=p|G|\)). Let \(G\) is a \(p\)-group of maximal class and order \(p^n\), \(n>3\). It is proved that \(|\Aut(G)|_p>|G|\), unless \(n=4\). All groups \(G\) of maximal class and order \(p^4\) with \(|\Aut(G)|_p=|G|\) are also listed. -- Of course, for \(p=2\), the results of this paragraph are known.