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We complete the classification of order $5$ nonsymplectic automorphisms onhyper-K\"ahler fourfolds deformation equivalent to the Hilbert square of a K3surface. We then compute the topological Lefschetz number of naturalautomorphisms of generalized Kummer fourfolds and we describe the geometry oftheir fix loci.

Some algebraic and geometric properties of hyper-K\"ahler fourfold symmetries