I have a $k$-variety $X$, and some algebraic group $G$ (not necessarily commutative) over $k$. Here, $k$ is a field of zero characteristic. I know that the functor $H^1(-,G)$ is contravariant, meaning that if I apply it to the morphism $X \to spec(k)$ I should get a morphism $H^1(k, G) \to H^1(X,G)$.
What is this morphism $H^1(k, G) \to H^1(X,G)$ explicitly? I.e. what does is send to what?
In terms of $G$-torsors it's just taking a product of a $G$-torsor over $k$ with $X$ -- the image of this map is the set of $G$-torsors over $X$ which is "trivial in the $X$-direction."