KSBA compactification of the moduli space of K3 surfaces with a purely non-symplectic automorphism of order four

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space K3 surfaces with purely non-symplectic automorphism order four and $U(2)\oplus D_4^{\oplus 2}$ lattice polarization. These can be realized as minimal resolution double cover $\mathbb {P}^{1}\times \mathbb {P}^{1}$ branched along specific $(4,\,4)$ curve. show that, up to finite group action, this is isomorphic Kirwan's partial desingularization GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ symmetric linearization.