Quotients of K3 surfaces modulo involutions

Let X be a K3 surface with an involution σ which has non-empty fixed locus X σ and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, σ) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index ≤ 2 or rational elliptic surfaces, and construct the only family of each of the three extremal case where X σ contains 10 (maximum possible) curves. We also classify rational log Enriques surfaces of index 2.