  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 $K3^{[2]}$-type admitting non symplectic involutions with
invariant lattices $U(2)\oplus D_4(-1)$ or $U(2)\oplus E_8(-2)$. This
complements the results obtained in [Mongardi and Wandel 2015], [Bossiere et al
2016], and the results from [arXiv:1603.00403] about the geometry of IHS
fourfolds constructed using the Hilbert scheme of $(1,1)$ conics on Verra
fourfolds. As a byproduct we find that IHS fourfolds of $K3^{[2]}$-type with
Picard lattice $U(2)\oplus E_8(-2)$ naturally contain non-nodal Enriques
surfaces.

Algebraic Geometry