I'll discuss recent work with Siad in which we compute the average number of 2-torsion ideal classes in pure cubic extensions over certain number fields. This has direct application to counting questions for certain types of S4/A4 quartic extensions. In the process of proving the result, we notice a phenomenon that leads us to predict an altered form of the Cohen-Lenstra heuristics for p-torsion classes in certain special families of number fields.