Double EPW sextics associated with Gushel–Mukai surfaces

Works by O’Grady allow to associate with a two-dimensional Gushel–Mukai (GM) variety, which is a K3 surface, a double Eisenbud–Popescu–Walter ( EPW ) sextic. We characterize the $K3$ surfaces whose associated double EPW sextic is smooth. As a consequence, we are able to produce symplectic actions on some families of smooth double EPW sextics which are hyper-Kähler manifolds. We also provide bounds for the automorphism group of GM varieties in dimension 2 and higher.