  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.
