11.126 (1990)

Do there exist a constant $h$ and a function $f$ with the following property: if a finite soluble group $G$ admits an automorphism $\phi$ of order 4 such that $|C_G(\phi)| \leqslant m$, then $G$ has a normal series $1 \leqslant M \leqslant N \leqslant G$ such that the index $|G : N|$ does not exceed $f(m)$, the group $N/M$ is nilpotent of class $\leqslant 2$, and the group $M$ is nilpotent of class $\leqslant h$?