Let $L/K$ be a Galois extension and let $X$ be a projective $L$-variety. Then, for every $\sigma \in $ Gal$(L,K)$, we get a projective $L$-variety $X^\sigma$ obtained via the action of $\sigma$ on the coefficients of the polynomials defining $X$ (considering $X$ as embedded in some projective space).
Can one define an action of Gal$(L/K)$ on the set of $L$-schemes X such that this action agrees with the usual action on varieties?