Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / a / v / Solution

For the trivial action, crossed homomorphisms $G\to \mathbb{Q}$ are ordinary group homomorphisms, while principal crossed homomorphisms vanish. The image of a finite group in the torsion-free additive group $\mathbb{Q}$ must be trivial, soDegree-zero cohomology is the invariant submodule; the action is trivial, so