Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of $G^*$. How many different can $(G\times G^*)$ admit a central extension by $T$? I'm wondering how explicitly one may express such groups.
EDIT: I originally claimed this was the semi-direct product that I was interested in. I regret the mistake.