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We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of  K 3 [2] -type admitting non-symplectic involutions with invariant lattices  U (2) ⊕  D  4 (−1) or  U (2) ⊕  E  8 (−2). This complements the results obtained in [ 43 ], [ 13 ], and the results from [ 29 ] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of  K 3 [2] -type with Picard lattice  U (2) ⊕  E  8 (−2) naturally contain non-nodal Enriques surfaces.

Verra Four-Folds, Twisted Sheaves, and the Last Involution