  We investigate the interplay between the moduli spaces of ample <2>-polarized
IHS manifolds of type K3^[2] and of IHS manifolds of type K3^[2] with a
nonsymplectic involution with invariant lattice of rank one. In particular we
geometrically describe some new involutions of the Hilbert square of a K3
surface, whose existence was proven in a previous work of Boissiere, Cattaneo,
Nieper-Wisskirchen and Sarti.
