multiple of a crossed homomorphism from finite group to a divisible one is principal

Question

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed homomorphism $f:\,\pi\rightarrow D$ (i.e., a function such that $f\left(\sigma\tau\right)=f\left(\sigma\right)^{\tau}+f\left(\tau\right)$ for every $\sigma,\tau\in\pi$ , where $\tau$ acts on $f\left(\sigma\right)$ ). Is it true that $nf$ is a principal homomorphism? Thanks in advance for any help or suggestion.

Answer 1

Actually this question was already asked in mathstack exchange. The answer is here first homology group with coefficients in divisible group