  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.
