Title: Symplectic action of groups of order four on K3^[2]-type manifolds. Abstract: When a group of order four G (either Z/4Z or (Z/2Z)^2) acts symplectically on a K3^[2]-type manifold X, then its action is always standard, meaning that we can deform the pair (X,G) to a pair (S^[2],G), where S^[2] is the Hilbert square of a K3 surface with a symplectic action of G, and the action of G on S^[2] is naturally induced. I will describe these two actions and construct for each one the general member of a projective family that admits it, starting from a family of K3 surfaces with a mixed (symplectic and non-symplectic) action of a group of order 4. Time permitting, I will also talk about the induced involutions on the Nikulin orbifold which is obtained by partial resolution of the quotient X/i, where i is a symplectic involution normal in G.