8.81 (1982)

Let $G$ be a finite $p$-group admitting an automorphism $\alpha$ of prime order $q$ with $|C_G(\alpha)| \leqslant n$.
$\qquad$ a) If $p = q$, then does $G$ have a nilpotent subgroup of class at most 2 and index bounded by a function of $n$?
$\qquad$ b) If $p \neq q$, then does $G$ have a nilpotent subgroup of class bounded by a function of $q$ and index bounded by a function of $n$ (and, possibly, $q$)?