  For the projective unitary group $PU_n$ with a maximal torus $T_{PU_n}$ and
Weyl group $W$, we show that the integral restriction homomorphism
  \[\rho_{PU_n} \colon H^*(BPU_n;\mathbb{Z})\rightarrow
H^*(BT_{PU_n};\mathbb{Z})^W\]
  to the integral invariants of the Weyl group action is onto. We also present
several rings naturally isomorphic to $H^*(BT_{PU_n};\mathbb{Z})^W$.
  In addition we give general sufficient conditions for the restriction
homomorphism $\rho_G$ to be onto for a connected compact Lie group $G$.
