On coincidence points of a function defined on a space with a finite abelian group action

Suppose a finite abelian group acts freely on a topological space X and f:X[implies]Y is a map. We use the Smith index and the Haefliger class to obtain results concerning the behavior of f with regard to the orbits of X. We prove an extension of a Borsuk-Ulam type theorem due to Munkholm.