I. J. Leary,
On The Integral Cohomology Of Wreath Products 

Under mild conditions on the space $X$, we describe the additive structure
of the integral cohomology of the space $X^p \times_{C_p}EC_p$ in terms
of the cohomology of $X$. We give weaker results for other similar spaces,
and deduce various corollaries concerning the cohomology of finite groups.