Hello. I have a question on covering spaces. Please let each space be paracompact, path-connected, locally path-connected and semi-locally simply connected.

Let $E\to B$ denote a normal covering space and $G$ its deck transformation group. It is the same as a $G$-principal bundle. On the other hand for a discrete group $G$ a $G$-principal bundle is a normal covering space.

$G$-principal bundles $E\to B$ are in one-to-one correspondence with pointed homotopy classes of maps $[B,BG]$. $BG$ denotes the classifying space construction.

$S^1$ is a classifying space for the integers. There is only one covering of $S^1$ with deck transformation group the integers, the universal covering $\mathbb{R}$. On the other hand $\pi_1(S^1)=[S^1,S^1]=\mathbb{Z}$. Can someone explain to me how this fits together?

If $G$ is discrete the set $[B,BG]$ should be in a one-to-one correspondence with normal $|G|$-sheeted coverings of $B$. Can one say that $[B,B\pi_1(X)]$ is in a one-to-one correspondence with all normal coverings of $B$?

Thanks.