Summary: This paper studies the effect, on the equivariant cohomology of a compact manifold \(X\) with a compact Lie group action \(G\), of an equivariant pair \(F = (f, \varphi) \) of maps, i.e., a smooth map \(f : X \to X\) and a Lie group homomorphism \(\varphi : G \to G\) such that \(f(gx) = \varphi (g) f(x)\). Such a pair mimics the Frobenius map for varieties in characteristic \(p\), and induces a graded map \(F^*\) on equivariant cohomology. Under certain transversality conditions (again mimicing the behaviour of the Frobenius map), we can define a ‘regularized’ Lefschetz number and prove a trace formula relating this number to local fixed point data. These fixed point data are extracted from the fixed-point groupoid associated to the pair \(F\). That is, \(F\) induces a functor of the groupoid defined by \(X\) and \(G\) and the fixed-point groupoid is the two-product of the graph of this functor and the diagonal functor. When the group action is trivial, this formula reduces to the usual Lefschetz trace formula for transverse maps.