Twisted Malle's Conjecture

In this paper, we use inductive methods similar to those employed in a 2025 paper by Alberts, Lemke Oliver, Wang and Wood in order to prove many new cases of the Twisted Malle’s Conjecture. Previously, this conjecture had only been proven for [Formula: see text] where [Formula: see text] is abelian, and for [Formula: see text]. We prove many new examples of this conjecture, such as for [Formula: see text] where [Formula: see text] is a finite abelian [Formula: see text]-group, [Formula: see text] is a [Formula: see text]-group, and [Formula: see text] is a proper abelian normal subgroup of B, and for [Formula: see text], where [Formula: see text] is odd and [Formula: see text] is a proper abelian normal subgroup of [Formula: see text].