K3 surfaces with a pair of commuting non-symplectic involutions

We study K3 surfaces with a pair of commuting involutions that are non-symplectic with respect to two anti-commuting complex structures that are determined by a hyper-K\"ahler metric. One motivation for this paper is the role of such $\mathbb{Z}^2_2$-actions for the construction of $G_2$-manifolds. We find a large class of smooth K3 surfaces with such pairs of involutions, but we also pay special attention to the case that the K3 surface has ADE-singularities. Therefore, we introduce a special class of non-symplectic involutions that are suitable for explicit calculations and find 320 examples of pairs of involutions that act on K3 surfaces with a great variety of singularities.