15.96 (2002)

An automorphism $\varphi$ of a group $G$ is called a splitting automorphism of order $n$ if $\varphi^n = 1$ and $x x^\varphi x^{\varphi^2} \dots x^{\varphi^{n-1}} = 1$ for any $x \in G$.
$\qquad$ a) Is it true that the derived length of a $d$-generated nilpotent $p$-group admitting a splitting automorphism of order $p^n$ is bounded by a function of $d$, $p$, and $n$? This is true for $n = 1$, see 7.53.
$\qquad$ b) The same question for $p^n = 4$.