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The mod-2 reduction is a functor from $\mathbb{Z}G$-modules to $\mathbb{F}_2G$-modules that preserves direct sums , because for instance it can be realized as tensoring with $\mathbb{F}_2$. Also if $M$...

I find Cauty's result a good crutch to think about ANRs: A metrizable space is an ANR if and only if every open subset has the homotopy type of a CW complex.

This is more of a long comment. I am not sure I understand your construction, but the sort of alternating sum you take has a preimage in the Burnside ring, and is often called the "Lefschetz invariant"...

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