Galois embedding problems and number field counting
Let $p$ be a prime number and $1\to H\to G\xrightarrow{q}T\to1$ be an exact sequence of finite groups, where $H$ is a nontrivial $p$-group. Fix a continuous surjection $\pi:\GQ\twoheadrightarrow T$ and finitely many $H$-conjugation-invariant local conditions $\cD$. Let $N_{G,\pi,\cD}(X)$ count the continuous surjections $\rho:\GQ\twoheadrightarrow G$ satisfying $q\circ\rho=\pi$ and these local conditions, with $|\Delta_{F_\rho}|\leq X$, where $F_\rho=\overline{\Q}^{\ker\rho}$. If at least one such lift exists, we prove that, for every $\varepsilon>0$, \[ X^{p/(|G|(p-1))} \ll_{G,\pi,\cD}…
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