On liftings of projective indecomposable G₍₁₎-modules

Let G be a simple simply connected algebraic group over an algebraically closed field k of characteristic p, with Frobenius kernel G₍₁₎. It is known that when p≥2h−2, where h is the Coxeter number of G, the projective indecomposable G₍₁₎-modules (PIMs) lift to G, and this has been conjectured to hold in all characteristics. In this paper, we prove that the PIMs lift to G if and only if they lift to G₍₁₎B. We also give examples of subgroup schemes G₍₁₎≤H≤G such that the PIMs can be lifted to H. Our work uses the group extension approach of Parshall and Scott, which builds on ideas due to Donkin, and we prove along the way various results about such extensions.