We study a variant of the inverse problem of Galois theory and Abhyankar's conjecture. If $p$ is an odd rational prime and $G$ is a finite $p$-group generated by $s$ elements, $s$ minimal, does there exist a normal extension $L/{\ma Q}$ such that $\hbox{\rm Gal }(L/{\ma Q})\cong G$ with at most $s$ rational primes that ramify in $L$? Given a nilpotent group of odd order $G$ with $s$ generators, we obtain a Galois extension $L/{\ma Q}$ with precisely $s$ prime divisors of ${\ma Q}$ ramified. Furthermore if $K$ is a number field satisfying $K\cap{\ma Q}(\zeta_{p_i^{n_i}})={\ma Q}$ for each rational prime $p_{i}$, such that $p_{i}^{n_i}|\circ (G)$, $p_{i}^{n_i+1}|\kern-.15cm / \circ (G)$, and such that there exists a rational prime $q$ inert in $K/{\ma Q}$, we obtain a Galois extension $E/K$ with precisely $s$ prime divisors of $K$ ramified. An adaptation of the Scholz-Reichardt method for the embedding problem is our main tool.