I am reading a book and there is the following statement:
Let $X$ be a nice enough space (e.g. a CW-complex) such that covering theory is applicable. Let $G$ be any group. Then one can identify the set $\mathrm{Hom}({\pi_1 (X, x_0}), G)$ with the set $Cov_G (X)$, which is the set of normal coverings of $X$ with covering group $G$.
I do not really understand this identification. I know about the classification of coverings, which says that for every subgroup of the fundamental group of $X$ one can find a covering space which has fundamental group the chosen subgroup. I do not see how this can be understood when one takes any homomorphism from the fundamental group to $G$. Any help is greatly appreciated.