14.96 (1999)

Suppose that a finite $p$-group $P$ admits an automorphism of order $p^n$ having exactly $p^m$ fixed points. By (E. I. Khukhro, Russ. Acad. Sci. Sbornik Math., 80 (1995), 435–444) then $P$ has a subgroup of index bounded in terms of $p$, $n$ and $m$ which is soluble of derived length bounded in terms of $p^n$. Is it true that $P$ has also a subgroup of index bounded in terms of $p$, $n$ and $m$ which is soluble of derived length bounded in terms of $m$? There are positive answers in the cases of $m = 1$ (S. McKay, Quart. J. Math. Oxford, Ser. (2), 38 (1987), 489–502; I. Kiming, Math. Scand., 62 (1988), 153–172) and $n = 1$ (Yu. A. Medvedev, see 10.68).