Invariant part of class groups and distribution of relative class group

We generalize the work of Roquette and Zassenhaus on the invariant part of the class groups to the relative class groups. Thus, we can show some statistical results as follows. For abelian extensions over a fixed number field K, we show infinite $C_p$-moments for the Sylow p-subgroup of the relative class group when p divides the degree of the extension. For sextic number fields with $A_4$-closure, we can show infinite $C_2$-moments for the Sylow 2-subgroup of the relative class group when the extensions run over a fixed Galois cubic field.