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 C 2 ≀ 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 = C 5 . We extend their results to prove that the average size of 3 -torsion in class groups of C 2 ≀ H -extensions is finite for any nilpotent group H .