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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.
