K3 surfaces with a pair of commuting non-symplectic involutions

We study K3 surfaces with a pair of commuting involutions that are nonsymplectic with respect to two anti-commuting complex structures that are determined by a hyper-Ka & BULL;hler metric. One motivation for this paper is the role of such Z22-actions for the construction of Riemannian manifolds with holonomy G2. We find a large class of smooth K3 surfaces with such pairs of involutions. After that, we turn our attention to the case that the K3 surface has ADE-singularities. We introduce a special class of nonsymplectic 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.