Cycle index generalises weight enumerator

With every linear code is associated a permutation group whose cycle index is the weight enumerator of the code (up to normalisation). There is a class of permutation groups (the IBIS groups) which includes the groups obtained from codes as above. With every IBIS group is associated a matroid; in the case of a code group, the matroid differs only trivially from that which arises from the code. In this case, the Tutte polynomial of the code specialises to the weight enumerator (by Greene’s Theorem), and hence also to the cycle index. However, in another subclass of IBIS groups, the base-transitive groups, the Tutte polynomial can be derived from the cycle index but not vice versa. I speculate that there might be an invariant which generalises both Tutte polynomial and cycle index.