  We describe the integral cohomology of $X/G$ where $X$ is a compact complex
manifold and $G$ a cyclic group of prime order with only isolated fixed points.
As a preliminary step, we investigate the integral cohomology of toric blow-ups
of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient
conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to
degenerate at the second page. As an application, we compute the
Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a
K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.
