Double EPW-sextics with actions of A7 and irrational GM threefolds

We construct two examples of projective hyper-Kähler fourfolds of K3[2]-type with an action of the alternating group A7, making them some of the most symmetric hyper-Kähler fourfolds. They are realized as so called double EPW sextics and this allows us to construct an explicit family of irrational Gushel-Mukai threefolds.