13.57 (1995)

Let $\varphi$ be an automorphism of prime order $p$ of a finite group $G$ such that $C_G(\varphi) \leqslant Z(G)$.
$\qquad$ a) Is $G$ soluble if $p = 3$? V. D. Mazurov and T. L. Nedorezov proved in (Algebra and Logic, 35, no. 6 (1996), 392–397) that the group $G$ is soluble for $p = 2$, and there are examples of unsoluble $G$ for all $p > 3$.
$\qquad$ b) If $G$ is soluble, is the derived length of $G$ bounded in terms of $p$?
$\qquad$ c) (V. K. Kharchenko). If $G$ is a $p$-group, is the derived length of $G$ bounded in terms of $p$? (V. V. Bludov produced a simple example showing that the nilpotency class cannot be bounded.)