use the following notation: $\underline{\, L}(G/H)=(X^H , L(G/H))$.

Let $\underline{L}$ be a $G$-equivariant local coefficient system of a finite $G$-CW-complex $X$. Recall that for a coefficient system $L$ on $X$, the group of $n$-cochains $\Gamma ^n(X; L)$ is defined as the group of all functions $c$ which take $n$-cells $\sigma $ in $X$ with characteristic map $h_{\sigma}:\Delta ^n\to X$ and send it to an element $c(\sigma )$ in $L(z_{\sigma })$ where $\Delta ^n$ is considered as the convex hull of a linearly independent subset $\{e_0,e_1,\dots e_n\}$ of ${\mathbb R}^{n+1}$ and $z_{\sigma }=h_{\sigma}(e_0)$. The coboundary operator $\delta:\Gamma ^{n-1}(X;L)\to \Gamma ^{n}(X;L)$ is defined as follows: For $c$ in $\Gamma ^{n-1}(X;L)$ and $\sigma $ a $n$-cell in $X$, we have $$(-1)^n(\delta c)(\sigma )=L(\gamma _\sigma )^{-1}c(\partial _0\sigma )+\sum _{i=1}^{n} (-1)^{i}c(\partial _i\sigma )\in L(z_{\sigma })$$ where $\partial _i\sigma $ is the $(n-1)$-cell with characteristic map $h_{\sigma}\circ d^n_i$ and $\gamma _\sigma (t)=h_{\sigma}((1-t)e_1+te_0)$. Here $d^n_i:\Delta ^{(n-1)}\to \Delta ^n$ is the affine map sending $e_j$ to $e_j$ when $j<i$ and to $e_{j+1}$ otherwise. Now for the $G$-equivariant coefficient system $\underline{L}$, we define $\Gamma ^n_G(X;\underline{L} )$ as elements in the direct sum $$\underline{\, c}=(c(G/H)) \in \bigoplus _{H\leq G}\Gamma ^n(X^H;L(G/H) )$$ which satisfy the following condition: $$c(G/H)(a\sigma )=\underline{\, L}(\hat{a})(z_{\sigma })(\, 