Exponents and the cohomology of finite groups

We will provide an example of a $p$-group $G$ which has elements of order $p^3$ in some of its integral cohomology groups but which also has the property that $p^2$ annihilates $\bar{H}^i(G;\mathbb{Z})$ for all sufficiently high $i$. This provides a counterexample to a conjecture of A. Adem which states that if a finite group $K$ has an element of order $p^n$ in one of its integral cohomology groups, then it has such an element in infinitely many of its cohomology groups.