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Let $\Omega\subseteq \mathbb{C}$ be the field of all constructible numbers (i.e. $\Omega$ is the smallest subfield of $\mathbb{C}$ which is closed under taking square roots). What is known about the automorphismgroup of the field $\Omega$?

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Every constructible field which is finite over Q has Galois group which is a 2-group. All finite 2-groups give constructible extensions of Q by a well-known theorem of Shafarevich. So the galois group you want is the inverse limit of all 2-groups.

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