Homology and Invariants commute

Question

Suppose we are given a differential graded vector space $(V=\bigoplus V_i,d)$ and representations of finite groups $G_i$ on $V_i$. Suppose the differential descends to $(V_i)^{G_i}\to (V_{i-1})^{G_{i-1}}$, where $W^G=\{w\in W\mid gw=w$ for all $g\in G\}$ the subspace of invariants.

This allows us to compute the homology before and after taking homology. I want to prove now $H(V)^G=H(V^G)$. I tried a lot but I am not sure if I have to add an assumption.

Answer 1

You need another assumption. A counterexample: let $V_1=V_0=\mathbb{F}$, else $V_i=0$, with $d:V_1 \to V_0$ the identity map. Then $H(V)=0$. Now choose group actions so that $V_1^{G_1}=0$ and $V_0^{G_0}=\mathbb{F}$.