Variations on a theme of Cline and Donkin

Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is obviously $G$-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups $G$ asserts that the (obviously $G$-stable) projective indecomposable modules (PIMs) $Q$ for the Frobenius kernels of $G$ have a $G$-module structure. It is sometimes just as useful (for a general module $Q$) to know that a finite direct sum $Q^{\oplus n}$ of $Q$ has a compatible $G$-module structure. In this paper, this property is called numerical stability. In recent work (arXiv:0909.5207v2), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of group schemes and a suitable version of $G$-stability, called strong $G$-stability. Among our results here is the presentation of a homological obstruction to the existence of a $G$-module structure, on strongly $G$-stable modules, and a tensor product approach to killing the obstruction.