orbit category which we will define now.

Let $G$ be a finite group and let $\Gamma $ denote the orbit category $\operatorname{Or}_G$. A functor from $\operatorname{Or}_G$ to the category of abelian groups ${\mathcal Ab}$ is called an ${\mathbb Z}{\Gamma}$-module. Morphisms between ${\mathbb Z}{\Gamma}$-modules are given by natural transformations. Given a $G$-CW-complex $X$, we define a chain complex of ${\mathbb Z}{\Gamma}$-modules by taking $C_n(X^?)$ as the functor $\operatorname{Or}_G\to {\mathcal Ab}$ which sends $G/H$ to the $n$-th cellular chains $C_n(X^H)$ and sends $\hat{a}:G/H\to G/K$ to the group homomorphism $a_*:C_n(X^K)\to C_n(X^H)$.

Let $\underline{L}$ be a $G$-equivariant local coefficient system of $X$. Suppose that there exists a ${\mathbb Z}{\Gamma}$-module $M$ such that $L(G/H)(x)=M(G/H)$ and $L(G/H)(\gamma )={\mathrm{id}}_{M(G/H)}$ for all $x\in X^H$ and all paths $\gamma $ in $X^H$. Then we have $$\Gamma ^n_G(X;\underline{L} )\cong \operatorname{Hom}_{{\mathbb Z}{\Gamma}}(C_n(X^?),M)$$ where the isomorphism is given by sending $\underline{\, c}=(c(G/H))$ in $\Gamma ^n_G(X; \underline{L} )$ to the homomorphism $\alpha : C_n(X^?) \to M$ defined by $\alpha (G/H)(\sigma )= c(G/H)(\sigma )$ for all $H\leq G$. The boundary maps at each $H$ are compatible with respect to inclusions and conjugations, so they combine together to give a ${\mathbb Z}{\Gamma}$-module map ${\partial}: C_n (X^?) \to C_{n-1} (X^{?})$ for every $n$. Using these boundary 