{\rm ind}_{\alpha}= {\rm ind}_{\alpha}\circ \partial_{n}^{H}$.]{}

3.  [For any group homomorphism $\beta: G\to K$ such that $\ker \beta\circ \alpha$ acts freely on $X$, one has $${\rm  ind}_{\alpha\circ \beta}= K^{{\rm geo}}_{K, j}(f_{1}\circ {\rm ind}_{\beta}\circ{\rm ind}_{\alpha}):K^{{\rm geo}}_{H, j}(X,A)\to K^{{\rm geo}}_{K, n}({\rm ind}_{\beta\circ \alpha}(X,A))$$ where $f_{1}: {\rm ind}_{\beta}{\rm ind}_{\alpha}\to {\rm ind}_{\beta\circ\alpha}$ is the canonical $G$-homeomorphism.]{}

4.  For any $j\in \mathbb{Z}$, any $g\in G$, the homomorphism

    $${\rm ind}_{c_(g):G\to G}K^{{\rm geo}}_{G, j}(X,A)\to K^{{\rm geo}}_{G, j}(\rm ind)_{c(g):G\to G}(X,A))$$

    agrees with the map $\mathcal{H}_{n}^{G}(f_{2})$, where $f_{2}: (X,A)\to {\rm ind}_{c(g):G\to G}(X,A)$ sends $x$ to $(1,g^{-1}x)$ and $c(g)$ is the conjugation isomorphism in $G$.

Given a geometric cycle for $(M, f, \sigma) $ for ${K^{\rm geo}_ {H}({X},{P})}$, form the cycle

$$({\rm ind}_{\alpha}M,{\rm ind}_{\alpha}f, \Sigma ),$$ where $\Sigma: {\rm  ind}_{\alpha}X \to  {\ensuremath{{\mathrm{Fred}}}}(P)$ is the map determined by $(g,x) \mapsto g \sigma(x)$. This is compatible with the equivalence relations defining the geometric $K$-homology groups. The homomorphism is seen to satisfy i-iv.

The spectral sequence
---------------------

In order to proof the coincidence of geometric twisted equivariant $K$-homology, we will introduce a spectral sequence based on $G$-equivariantly contractible covers. This is the dual (homological instead of cohomological) counterpart of the spectral sequence constructed in [@barcenasespinozauribevelasquez] for twisted equivariant $K$-theory. Due to the poor properties 