Are all topological group structures on the Baire Space of the form $G^{\mathbb N}$ for $G$ a countable discrete group?

Question

This question was accidentally trivial: For any countably infinite discrete group $G$ we have that $G^{\mathbb N}$ is a topological group structure on the Baire Space, as pointed in Qiaochu Yuan's answer. What about the coverse? Are all topological group structures on the Baire space of this form?

Answer 1

It's still pretty straightforward: we can take any countable product $\prod_{i=1}^{\infty} G_i$ where $G_i$ is a sequence of countable discrete groups.

I suppose there's some work to do to show that there exists such a group that cannot be isomorphic to a group of the form $G^{\mathbb{N}}$.