Let $X$ and $Y$ be K3 surfaces over the complex numbers.


Under what assumptions, do there exist 


1. a finite group $G\_X$
2. a finite group $G\_Y$
3. a $G\_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology)
4. a $G\_Y$-gerbe $\mathcal{Y}\to Y$
5. an isomorphism $\mathcal{X}\cong \mathcal{Y}$ (of algebraic stacks)?



Common gerbes over two K3 surfaces
As Jason Starr says in his comments, such data exists if and only if $X$ is isomorphic to $Y$. 


Indeed, let $\mathcal{X}\to X$ be a $G\_X$-gerbe, and let $\mathcal{Y}\to Y$ be a $G\_Y$-gerbe. As the (abstract) groups $G\_X$ and $G\_Y$ are finite, the stacks $\mathcal{X}$ and $\mathcal{Y}$ are finite type separated DM stacks. 
Since $\mathcal{X}$ and $\mathcal{Y}$ are isomorphic, the coarse space of $\mathcal{X}$ is isomorphic to the coarse space of $\mathcal{Y}$. We are now done by the following lemma.


Lemma 1. The coarse space of $\mathcal{X}$ is $X$ and the coarse space of $\mathcal{Y}$ is $Y$. 


