On projective K3 surfaces $\mathcal{X}$ with $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{X})=(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$

We prove that every K3 surface with automorphism group $(\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^2$ admits an explicit birational model as a double sextic surface. This model is canonical for Picard number greater than 10. For Picard number greater than 9, the K3 surfaces in question possess a second birational model, in the form of a projective quartic hypersurface, generalizing the Inose quartic.