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The non-action part For a given finite group G, is there a cover of regular Inverse GaloisP^1 over Q s.t. over C it's G-Galois?
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The non-action part of regular Inverse Galois

For any finite group, G, we can find a cover of ℙ1ℂ which is G-Galois. The regular inverse Galois problem is equivalent to there existing such a cover that descends with action to ℚ. My question is about the easier problem: given a finite group G, can we find a cover of ℙ1ℂ such that it descends to ℚ as a mere cover (meaning not necessarily with group action)?

From the results that I know, I would be really surprised if this is solved. But what is known? And where is it written?