During some exercises I came upon the following questions that I cannot answer myself. Given a topological group $G$ we can build the completion using cauchy sequences and denote this completion by $\bar{G}$. This is again a topological group. Let $\phi : G \rightarrow \bar{G}$ be the usual map sending each element of $G$ to the equivalence class of the constant sequence. I know (not hard to show) that $\phi$ is injective if and only if $G$ is hausdorff. My questions are as follows: (EDIT: Assuming $G$ is abelian and satisfies the first axiom of countability)

If $G$ is hausdorff, is $\bar{G}$ hausdorff? Or is $\bar{G}$ automatically hausdorff? (Elements that cannot be seperated in $G$ are mapped to the same equivalence class)

When is $\bar{G}$ isomorphic to $\bar{\bar{G}}$ and how can I show that?

I'd also appreciate it, if someone could point me to some good literature.

Best regards