Let $X$ be an algebraic variety (say, projective, irreducible and smooth), defined over a field $K$, and let $L$ be a Galois extension. I am interested in algebraic varieties $Y$, defined over $K$, such that there exists an isomorphism $\psi\colon X_L\to Y_L$ defined over $L$ (but not over $K$ in general).

The map $\psi$ induces a 1-co cycle from $Gal(L/K)$ to the group $Aut(X_L) $ of $L$-automorphisms, and it is easy to check that the image in $H^1 (Gal, Aut(X_L))$ only depends on the class of $Y$, up to $K$-isomorphisms. My question is the following: is every element of $H^1$ obtained? In other words, we have a map from the set of $L$-forms of $X$ to $H^1 (..)$, the map is injective but is it surjective?

If no, could it be true in some natural cases, if yes I would be happy to see a reference.

algebraic space$X$ such that $X_L$ is isomorphic to $Y$, giving rise to the given cocycle. Mathieu Huruguen has given explicit examples of (non-quasiprojective) toric varieties $Y$ for which this phenomenon happens. – ACL Mar 1 at 21:28