On Wohlfahrt series and wreath products

Suppose that a group A contains only a finite number of subgroups of index d for each positive integer d. Let G≀Sn be the wreath product of a finite group G with the symmetric group Sn on {1,…,n}. For each positive integer n, let Kn be a subgroup of G≀Sn containing the commutator subgroup of G≀Sn. If the sequence satisfies a certain compatible condition, then the exponential generating function of the sequence takes the form of a sum of exponential functions.