A minkowski-style bound for the orders of the finite subgroups of the Cremona group of rank 2 over an arbitrary field

Let $\mathrm{Cr}(k)=\operatorname{Aut}k(X,Y)$ be the Cremona group of rank 2 over a field $k$. We give a sharp multiplicative bound $M(k)$ for the orders of the finite subgroups $A$ of $\mathrm{Cr}(k)$ such that $|A|$ is prime to $\mathrm{char}(k)$. For instance $M(\mathbf Q)=120960$, $M(\mathbf F_2)=945$ and $M(\mathbf F_7)=847065600$.