Defined as follows. On a given degree $n$, $C_n({\mathcal{N}_{G} \mathcal{U}}, M)$ is the $R$-module $$\underset{U_{\sigma}= U_{\sigma_{1}} \cap \ldots \cap U_{\sigma_n} \in {\mathcal{N}_{G} \mathcal{U}}^{n}}\bigoplus M(U_\sigma)\otimes_R  R[U_\sigma],$$ where $R[U_\sigma]$ denotes a free $R$ module of rank one and $U_\sigma $ denotes a non empty length $n$ intersection.

The boundary maps of the simplicial set ${\mathcal{N}_{G} \mathcal{U}}$ tensored with the $R$-module homomorphisms given by the functoriality of $M$ give the differential

$$\begin{gathered}
d_n:  \underset{U_{\sigma}= U_{\sigma_{1}} \cap \ldots \cap U_{\sigma_{n+1}} \in {\mathcal{N}_{G} \mathcal{U}}^{n}}\bigoplus M(U_\sigma)\otimes_R  R[U_\sigma] \\ \longrightarrow \underset{U_{\tau}= U_{\tau_{1}} \cap \ldots \cap U_{\tau_{n}} \in {\mathcal{N}_{G} \mathcal{U}}^{n-1}}\bigoplus M(U_\tau)\otimes_R  R[U_\tau].\end{gathered}$$

We will analyze now the coefficient system associated to twisted equivariant $K$-homology.

Let $X$ be a space satisfying condition \[conditionCMP\]. $i_\sigma: G/H_\sigma\to  U_\sigma \to X$ be the inclusion of a proper $G$-orbit into the contractible slice $U_\sigma \subset X$ and consider the Borel cohomology group $H^3( EG\times _G G/H_\sigma, {{\mathbb Z}})$. Given a class $ P \in  H^3(EG \times_G X, {{\mathbb Z}})$, we will denote by $\widetilde{H_{P_\sigma}}$ the central extension $1\to S^1 \to \widetilde{H_{P_\sigma}} \to  H_\sigma \to  1 $ associated to the class given by the image of $P$ under the maps $$\omega_\sigma : H^3(EG\times X, {{\mathbb Z}})\overset{i_\sigma ^* }{\to} H^3(EG\times_G G/H_\sigma, {{\mathbb Z}})\overset{\cong}{\to} H^3(BH_\sigma, {{\mathbb 