Let $G$ be a countable discrete group. A group compactification of $G$ is a compact Hausdorff topological group $H$ such that there is a group homomorphism $\iota\colon G\to H$ with dense image. For example, for every group $G$ there is the Bohr compactification, which is in some sense the largest one. But for some reasons, I am looking for metrisable compactifications, equivalently, for second countable compactifications. Which discrete countable groups have non-trivial largest metrisable compactifications? And a follow up question: For which groups we can also demand that $\iota$ is injective? I found here on MO that a finitely generated group admits an injective homomorphism to a compact group if, and only if, it is residually finite. It is not hard to see that all residually finite groups embbed injectively in a metrisable compactification. So are there any other examples of such countable discrete groups embedded injectively into compact metrisable groups?