I would like to understand the concept of classification in the context finite groups. For finite abelian groups and finite simple groups, it's clear to me what is meant by classification. However, these are the cases where a "perfect classification" turns out to work fine.
Because I have a pretty good understanding of nilpotent groups, I asked myself how the generalization of the classification of finite abelian groups to finite nilpotent groups would look like. In a certain sense, the statement "Every finite nilpotent group is the direct product of p-groups" contains the classification of finite abelian groups as a special case. So at least the classification of finite nilpotent groups is now reduced to the classification of finite (indecomposable) p-groups. But even so the structure of p-groups is "relatively simple" and well understood, trying to enumerate the isomorphism classes is challenging and doesn't seem to add much value.
Would an acceptable solution for the classification problem of finite nilpotent groups necessarily include the possibility for an enumeration of the isomorphism classes? Or are "more practical" solutions also acceptable? For example, would it be sufficient to give both an "efficient" algorithm for constructing at least one representative of each isomorphism class of a given order, and an "efficient" algorithm for deciding whether two representatives corresponds to the same isomorphism class? What would be the role/importance (with respect to the classification problem) of characteristic numbers that only depend on the isomorphism class?