We describe an inductive method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group G. Our method relies on having asymptotic counts for certain T-extensions for some normal subgroup T of G, uniform bounds for the number of such T-extensions, and possibly weak bounds on the asymptotic number of G/T-extensions. However, we do not require that most T-extensions of a G/T-extension are G-extensions, i.e. we are not just counting wreath products. Our new results use T either abelian or S_m^3, though our framework is general. This talk is on joint work with Brandon Alebrts, Robert Lemke Oliver, and Jiuya Wang.