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 $T \trianglelefteq G$ where $T$ is abelian, and for $S_3^m \trianglelefteq S_3 \wr B$. We prove many new examples of this conjecture, such as for $A \wr M \trianglelefteq A \wr B$ where $A$ is a finite abelian $p$-group, $B$ is a $p$-group, and $M$ is a proper abelian normal subgroup of $B$, and for $C_p \wr M \trianglelefteq C_p \wr B$, where $|B|$ is odd and $M$ is a proper abelian normal subgroup of $B$.