Small-Subgroup Criteria for Liftability of Automorphism Groups of Smooth Hypersurfaces

In this paper, building on our previous Sylow criteria, we establish small-subgroup criteria of liftability and $F$-liftability for the linear automorphism group $G$ of smooth hypersurfaces $X$ over algebraically closed field of characteristic zero. When $\mathrm{d}\mathrm{i}\mathrm{m}X=p-2$ for some odd prime $p$, we prove that ($F$-)liftability of any finite subgroup of $G$ can be tested on its $p$-subgroups of order at most $p^2$. When $X$ is a degree $p$ hypersurface of dimension $2p-2$, we prove that ($F$-)liftability of $G$ can be tested on all its $p$-subgroups of order at most $p^3$, which is sharp for $p\ge 5$. When $p=3$, this bound improves to $9$, giving the corresponding criteria for smooth cubic fourfolds.