Assume G is discrete. I am trying to determine the the bijection {based regular G-covers $(E,e) \to (B,b)$} and {homomorphisms $\pi_1(B,b) \to G$} so that I can prove, for homework, that if no non-zero $\pi_1(B,b) \to G$ exists, then $E \cong B \times G$.
I have shown that regular G-covers have the property where the deck transformations act transitively on the fibers of the basepoint of $(B,b)$.