Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ corresponding to $F^{'}$ .
I know that this covering space should be free because any subgroup of free groups is also free.because of commutators I have a feeling that this covering space most be 
but I don't know my Idea is right or wrong?please help me with your knowledge,thank you very much.
Edit:there is something that make me a little confused,for $S^{1} \vee S^{1}$ the universal covering space $\widetilde{X}$ is 
and if we take $\frac{\widetilde{X}}{F^{'}}$, this is abelian covering space of $F$,is it true that $\pi_{1}(\frac{\widetilde{X}}{F^{'}})=F^{'}$?