Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X_K$ be an étale $G$-torsor.

For what field extensions $K/k$ does any such $Y$ descend to $k$? I.e.: for what field extensions $K/k$ does there exist $Y_0\to X$ an étale $G$-torsor, such that $(Y_0)_K\cong Y$ and $(Y_0)_K\to X_K$ is the map $Y\to X_K$?

Equivalently, for what field extensions $K/k$ is the map $H^1(X,G)\to H^1(X_K, G)$ surjective?

For example, if $k$ is algebraically closed, for any extension $K/k$ of algebraically closed fields the map is an isomorphism, by the smooth base change theorem.

**Example** Let $k$ be the maximal unramified extension of $\mathbf{Q}_p$. Does any finite field extension $K/k$ satisfy this property?