  The period morphism of polarized hyper-K\"ahler manifolds of K3$^{[m]}$-type
gives an embedding of each connected component of the moduli space of polarized
hyper-K\"ahler manifolds of K3$^{[m]}$-type into their period space, which is
the quotient of a Hermitian symmetric domain by an arithmetic group. Following
work of Stellari and Gritsenko-Hulek-Sankaran, we study the ramification of
covering maps between these period spaces that arise from the action of some
groups of isometries.
