Symplectic actions of groups of order 4 on K3^[2]-type manifolds, and standard involutions on Nikulin-type orbifolds

Abstract

Given a K3^[2]-type manifold X with a symplectic involution i, the quotient X/i admits a Nikulin orbifold Y as terminalization. We study the symplectic action of a group G of order 4 on X, such that i belongs to G, and the natural involution induced on Y (the two groups give two different results). We give a lattice-theoretic classification of X and Y in the projective case, and give some explicit examples of models of X. We also give lattice-theoretic criteria that a Nikulin-type orbifold N has to satisfy to admit a symplectic involution that deforms to an induced one.