What is the Galois group of the Hyperreal Numbers?

The Galois Group of $$\mathbb{R}$$ as an extension of $$\mathbb{Q}$$ is trivial. That’s because any field automorphism of $$\mathbb{R}$$ is order preserving, so since $$\mathbb{Q}$$ is dense in $$\mathbb{R}$$, it follows that the only field automorphism of $$\mathbb{R}$$ that fixes $$\mathbb{Q}$$ is the identity.

But I don’t think $$\mathbb{Q}$$ is dense in the field of hyperreal numbers. So my question is, what is the Galois group of the field of hyperreal numbers as an extension of $$\mathbb{Q}$$?

I expect it to be a pretty large group, maybe something that involves a Hamel basis of $$\mathbb{R}$$ over $$\mathbb{Q}$$.