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It is known that the inverse Galois problem for solvable groups was solved by Shafarevich. My question is the following: given $G$ a finite $2$-group and $s$ an element of order $2$ in $G$. Can we find a Galois extension $K$ of $\mathbb{Q}$ with Galois group $G$ such that $s$ "is" the complex conjugation on $K$?

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