Take X to be a scheme over a (say algebraic closed) field k and G a group scheme act on X. I want to get more convenient knowledge about G-invariant vector bundles over X.

I'm inspired by this example: $X=A^n$ is the affine space and $G=G_m$ is the multiplication group which acts on X through obvious scaler multiplication. Then to know a vector bundle over X invariant under G is equivalent to know a vector bundle over the projective space $P^{n-1}$.

Of course I do not expect in general there exists a scheme Y playing the role as $P^{n-1}$ in the above example, but under what conditions about the action of G can we get such a scheme? More weakly, if we allow Y to be an object in some bigger category (algebraic space, stack, presheaf...), can we get similar results which adapt to more general actions?

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edit: i should have used the term G-equivariant...sorry