  Let X be a topological Hausdorff space together with a continuous action of a
finite group G. Let R be the ring of integers of a number field F. Let E be a
G-sheaf of flat R-modules over X and let $\Phi$ be a G-stable paracompactifying
family of supports on X. We show that under some natural cohomological
finiteness conditions the Lefschetz number of the action of g in G on the
cohomology $H_\Phi(X,E) \otimes_{R} F$ equals the Lefschetz number of the
g-action on $H_\Phi(X^g, E_{|X^g}) \otimes_{R} F$, where $X^g$ is the set of
fixed points of g in X. More generally, the class $\sum_j (-1)^j [H^j_\Phi
(X,E) \otimes_R F]$ in the character group equals a sum of representations
induced from irreducible F-rational representations $V_\lambda$ of $H$ where
$H$ runs in the set of G-conjugacy classes of subgroups of G. The integral
coefficients $m_\lambda$ in this sum are explicitly determined.
