Im trying to find a way to connect a possible solution of the inverse Galois problem on simple groups to a more general solution on any finite group.

I've tryied to mess with the embedding problem for a bit but with no success, any help?

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    $\begingroup$ This seems more like a search for collaborators than the kind of focussed question that belongs on MO. $\endgroup$ – LSpice Jun 15 at 16:54
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    $\begingroup$ Not an expert, just going by rough memory here: Since the inverse Galois problem for finite simple groups is almost solved (only very few, maybe two or three cases are left if I remember correctly), I wouldn't have too much hope for such a result. If that was easily done, it would solve the inverse Galois problem (almost) completely. So what you're asking about is more or less a full solution to the inverse Galois problem. $\endgroup$ – Johannes Hahn Jun 15 at 16:56
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    $\begingroup$ Also, the inverse Galois problem for abelian groups is trivial, but for solvable groups IIRC it's a very hard result by Šafarevič. $\endgroup$ – Gro-Tsen Jun 15 at 17:09
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    $\begingroup$ @JohannesHahn The inverse Galois problem is solved for almost all the sporadic simple groups. If memory serves it is still far from solved for all Chevalley groups, and those over the field of $p^n$ elements for $n>1$ tend to be particularly hard. $\endgroup$ – Noam D. Elkies Jun 15 at 17:14
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    $\begingroup$ I believe that $M_{23}$ is the only outstanding unknown sporadic example, which seems strange, but apparently is just happens to be the single examples that evades all of the known techniques. $\endgroup$ – Derek Holt Jun 15 at 18:33

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