On the average size of $3$-torsion in class groups of $C_2 H$-extensions

The Cohen-Lenstra-Martinet heuristics lead one to conjecture that the average size of the p-torsion in class groups of G-extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of $p = 3$ for permutation groups G of the form C₂ H for a broad family of permutation groups H, including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as H = C₅. We extend their results to prove that the average size of $3$-torsion in class groups of C₂ H-extensions is finite for any nilpotent group H.