I previously mentioned a problem concerning polynomials, whose motivation came from thinking about weight one forms and the inverse Galois problem for finite subgroups of \(\mathrm{GL}_2(\mathbf{C}).\) I still like the polynomial problem, but I realized that I was confused about the intended application. Namely, given a weight one form with projective image \(A_5,\) there is certainly a unique minimal lift up to twist, but the images of the twists also automatically have image given by a central extension \(A_5.\) So, just by twisting, one can generate all such groups as Galois groups by starting with a minimal lift. More prosaically, every central extension of \(A_5\) by a cyclic group is either a quotient of \(A_5 \times \mathbf{Z}\) or of \(\widetilde{A}_5 \times \mathbf{Z}\) where \(\widetilde{A}_5\) is the Darstellungsgruppe of \(A_5\) (which is \(\mathrm{SL}_2(\mathbf{F}_5)).\) So, to solve the inverse Galois problem for central extensions of \(A_5\), it suffices to solve it for \(\mathrm{SL}_2(\mathbf{F}_5).\) That is not entirely trivial, but it is true.
I still think it’s an interesting problem to determine which extensions of \(A_5\) by cyclic groups occur as the Galois groups of minimally ramified up to twist extensions, but that is not the same as the inverse Galois problem.