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.Furthermore, if $G$ is a topological group, we equip $\tilde{G}$ with the compact open topology.

Assume that $G$ is a locally compact abelian group. Is true to say $\tilde{G}$ is a locally compact abelian topological group, too? Under which condition do we have the duality $$ \tilde{\tilde{G}}\sim G$$

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