Fiber bundle over $SL_2(\mathbb C)/\mathbb C^*$

Question

Let $F:=N/\Gamma$ where $\Gamma$ is discrete be a compact nilmanifold. I want to construct a fiber bundle over $SL_2(\mathbb C)/\mathbb C^*$ with fiber $N/\Gamma$.

1. I have read somewhere that if one considers the homomorphism $\varphi:\mathbb C^* \rightarrow \mathrm{Aut}(F)$ then $SL_2(\mathbb C)\times_\varphi F $ is an $F$-fiber bundle over $SL_2(\mathbb C)/\mathbb C^*$. Can one explain this construction to me?

2. Can we show that $\varphi (\mathbb C^*)$ is central subgroup of the nilpotent group $N$?

3. Consider $\varphi|_{\mathbb S^1}$. What will the new fiber bundle be?