constructing principal $G$-bundles over a topological space

Question

Let $G$ be an abelian Lie group, and let $X$ be a topological space. I know that principal $G$-bundles over $X$ can be classified by the cohomology group $\check{H}^1(X,G)$. Since $G$ is abelian, $\check{H}^1(X,G)$ is isomorphic to $H^2(X,G)$. For a given non-trivial principal $G$-bundle $\pi:P \to X$, If $H^2(X,G)$ be isomorphic to $Z$, is it possible to construct other principal $G$-bundles over $X$ from $\pi$? If yes, how can I do this?