Let $\mathsf{k}$ be a field of characteristic $0$, and consider $\mathsf{k}(x,y)$.

If $\mathsf{k}$ is algebraically closed, then every field $L$ such that the inclusion $\mathsf{k} \subset L \subset \mathsf{K}(x,y)$ holds is a purely transcendental extension of the base field (i.e., Castelnuovo's Theorem implies a positive solution to the Lüroth problem in two dimensions).

Now suppose that $\mathsf{k}$ is **not algebraically closed**.

Question: can we have a finite group $G$ of field automorphisms of $\mathsf{k}(x,y)$, fixing $\mathsf{k}$, such that $\mathsf{k}(x,y)^G$ is not a purely transcendental extension of $\mathsf{k}$?

I am looking for an explicit example of $G$ such that $\mathsf{k}(x,y)^G$ is not rational.