Lat $\mathbf {Z}_p$ denote the ring of $p$-adic integers. Let $W\subset \mathrm {GL}(n,\mathbf {Z}_p)$ be a finite group such that $p$ does not divide the order of $W$. The group $W$ acts on $K((\mathbf {Z}_p)^n,2)$. Let $X(W,p,n)_p$ be the $p$-completion of the space $K((\mathbf {Z}_p)^n,2)\times _W EW$. We ify homotopy es of maps between spaces $X(W,p,n)_p$.