The lifting–Oort-group conjecture for automorphism groups of cyclic curves

Let $X$ be a smooth cyclic curve over a field $k$ of characteristic $p$. Let $N$ be a cyclic normal subgroup of $G=\operatorname{Aut}_{k}(X)$ such that the quotient $X/N$ has genus $0$, and let $n$ be the order of $N$, where $(2n,p)=1$. A finite group is an Oort group for $k$ if every faithful action of it on a smooth projective curve over $k$ lifts to characteristic $0$. Lifting–Oort-group conjecture. The group $G$ is liftable to characteristic $0$ if and only if $G$ is an Oort group for $k$. This proposes a general criterion for lifting automorphism groups of cyclic curves, extending the preceding result for $p>2g+1$; the supplied text gives no resolution of the conjecture.