Restrictions to G( p) and G(r) of rational G-modules

We fix a prime p and consider a connected reductive algebraic group G over a perfect field k which is defined over p. Let M be a finite-dimensional rational G-module M, a comodule for k[G]. We seek to somewhat unravel the relationship between the restriction of M to the finite Chevalley subgroup G( p)⊂G and the family of restrictions of M to Frobenius kernels G(r) ⊂G. In particular, we confront the conundrum that if M is the Frobenius twist of a rational G-module N,M=N(1), then the restrictions of M and N to G( p) are equal whereas the restriction of M to G(1) is trivial. Our analysis enables us to compare support varieties (and the finer non-maximal support varieties) for G( p) and G(r) of a rational G-module M where the choice of r depends explicitly on M.