If Y is a homotopy representation of the finite group G of order n and X is a finite G-CW complex such that, for each subgroup H of G, H ∗ ( X H ; Z n ) = H ∗ ( Y H ; Z n ) {H_ \ast }({X^H};{\mathbb {Z}_n}) = {H_ \ast }({Y^H};{\mathbb {Z}_n}) then there exists a G-map f : X → Y f:X \to Y such that f ∗ H : H ∗ ( X H ; Z n ) → H ∗ ( Y H ; Z n ) f_ \ast ^H:{H_ \ast }({X^H};{\mathbb {Z}_n}) \to {H_ \ast }({Y^H};{\mathbb {Z}_n}) is an isomorphism for each subgroup H.