On the nonsymplectic involutions of the Hilbert square of a K3 surface

We investigate the interplay between the moduli spaces of ample $\langle 2\rangle$-polarized IHS manifolds of type $\mathrm{K3}^{[2]}$ and of IHS manifolds of type $\mathrm{K3}^{[2]}$ with a non-symplectic involution with invariant lattice of rank one. In particular, we describe geometrically some new involutions of the Hilbert square of a K3 surface whose existence was proven in a previous paper of Boissière, Cattaneo, Nieper-Wisskirchen, and Sarti.