Let $G$ be a Lie group and $M$ a smooth manifold. The universal cover $\tilde M$ is a principal $\pi_1(M)$-bundle over $M$.
Question: Why does any homomorphism $\phi:\pi_1(M)\to G$ induces a principal $G$-bundle over $M$?
A priori I thought of extendig the cocycles $H_{\alpha\beta}:\pi_1(M)\times U_{\alpha\beta}\to\pi_1(M)$ associated to the bundle $\tilde M$ with the morphism $\phi$, but in general it is not possible to extend group homomorphisms, so I don't see any obvious way to do it.
My guess would be taking the space $\tilde M\times G$ on which $\pi_1(M)$ acts and then thaking the quotient, but I don't see immediately if this work.