Coincidences for maps of spaces with finite group actions

Let G be a finite group acting freely in a Hausdorff, paracompact, connected and locally pathwise connected topological space X such that Hi(X,Z)=0 for 0<i<m and Hm+1(G,Z)≠0. Let f :X→Y be a map of X to a finite k-dimensional CW-complex Y. We show that if m⩾|G|k, then f has a (H,G)-coincidence point for some nontrivial subgroup H of G.