  We propose new tools based on basic lattice theory to calculate the integral
cohomology of the quotient of a manifold by an automorphism group of prime
order. As examples of applications, we provide the Beauville--Bogomolov forms
of some irreducible symplectic orbifolds; we also show a new expression for a
basis of the integral cohomology of a Hilbert scheme of two points on a
surface.
