There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper, we give sufficient conditions that a given finite group [Formula: see text] occurs infinitely often as a Galois group over the rationals [Formula: see text] with all nontrivial inertia groups of order [Formula: see text]. Notably any such realization of [Formula: see text] can be translated up to a quadratic field over which the corresponding realization of [Formula: see text] is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group [Formula: see text] — if such a polynomial is available — and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, the applications to [Formula: see text] and [Formula: see text] are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for [Formula: see text] by the first author in [J. König, On intersective polynomials with nonsolvable Galois group, Comm. Alg. 46(6) (2018) 2405–2416.