The following is a related result of Bredon et al and Lowen. You may find more in [this book][1] by Bredon.

Theorem. Let $G$ be a totally disconnected compact group that acts on a locally compact Hausdorff space $X$, and let $k$ be a field of characteristic $0$. Then the orbit projection $X\rightarrow X/G$ induces an isomorphism $$H_c^{*}(X/G;k)\cong \text{Fix}(G;H_c^{*}(X;k)).$$

Also, take a look at [this paper][2] by Satea Deo. [1]: http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf [2]: http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a7b_1319.pdf

The following is a related result of Bredon et al and Lowen. You may find more in this book by Bredon.

Theorem. Let $G$ be a totally disconnected compact group that acts on a locally compact Hausdorff space $X$, and let $k$ be a field of characteristic $0$. Then the orbit projection $X\rightarrow X/G$ induces an isomorphism $$H_c^{*}(X/G;k)\cong \text{Fix}(G;H_c^{*}(X;k)).$$

Also, take a look at this paper by Satea Deo.