Are these maps group homomorphisms?

Question

Let $G$ be a finite group, $M$ a trivial $G$-module and let $f:G \times G \to M$ be a 2-cocycle.

Question: Are the following maps $f_1,f_2$ group homomorphisms ?

$$f_1: G \to M,\; g \mapsto \sum_{i=1}^{|G|}f(g,g^i)$$ $$f_2: G \to M,\; g \mapsto \sum_{h\in G}f(g,h)$$

Answer 1

Of course, no. Take an arbitrary map $\alpha:G\to M$. Then $f(g,h)=\alpha(g)-\alpha(gh)+\alpha(h)$ is a cocycle which does not satisfy your conditions.