A note on \(p'\)-automorphism of \(p\)-groups \(P\) of maximal class centralizing the center of \(P\) (Q1355565)

Let \(P\) be a \(p\)-group of maximal class and of order \(p^n\), \(n\geq 4\), \(p\) odd. The author proves the following about a Hall \(p'\)-subgroup of \(\Aut(P)\): 1) \(C_H(Z(P))\) is cyclic of order dividing \(p-1\). If, moreover, \(|H|=(p-1)^2\), then \(C_H(Z(P))\) has order \(p-1\). 2) If \(H\) is a Sylow \(q\)-subgroup of \(\Aut(P)\) with \(q\) odd and \(q\) dividing \(p-1\), then \(C_H(Z(P))\) acts regularly on \(P/Z(P)\) if and only if \(|P|\leq p^{q+1}\).