  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 $p\ge 2h-2$, 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 explore the
lifting problem via extensions of algebraic groups, following the work of
Parshall and Scott who in turn build upon ideas due to Donkin. We prove various
results which augment this approach, and as an application demonstrate that the
PIMs lift to $G_{(1)}H$, for particular closed subgroups $H \le G$ which
contain a maximal torus of $G$.
