Let $G$ be a group. We define $$\tilde{G}=\{\phi:G \to \mathbb{T}\mid \phi(gh){\phi(g)}^{-1}{\phi(h)}^{-1}\in Tor(\mathbb{T})\}$$

where $Tor(\mathbb{T})$ is the group of torsion elements of the unit circle $\mathbb{T}$.

Then $\tilde{G}$ is a group with the obvious operation.

When $G$ is a locally compact abelian group, is $\tilde{G}$ a locally compact abelian group with the compact open topology? Under which condition do we have the duality $$ \tilde{\tilde{G}}\sim G$$

Is there a notion or terminology for this $\tilde{G}$?