Showing that every element in G can be written a certain way

Let G be a finite group which possesses an automorphism ##\sigma## that has no nontrivial fixed points and ##\sigma ^2## is the identity map. Prove that ##G## is abelian.

So there's a hint, and it tells me first to established that every element in ##G## can be written as ##x^{-1} \sigma (x)## for some ##x \in G##. I have two questions. Why does showing that ##f(x) x^{-1} \sigma (x)## is injective prove the hint, and how could I ever approach this problem without knowing the hint?