Fix a prime $p$. Then the question is about the difficulty of classifying finite $p$-groups. Originally I was going to ask if this was a "wild" problem but thanks to Joel David Hamkins' answer to question
<a href="http://mathoverflow.net/questions/10481/when-is-a-classification-problem-wild">10481</a> I now know to ask if it is a smooth problem.

At one time there was a minor industry producing papers which showed that a particular list of invariants did not classify finite $p$-groups so my understanding is that it would be remarkable if this problem was smooth.

Also for each prime $p$ and each $n$ the number of groups of order $p^n$ (up to isomorphism) is finite. This gives a sequence of integers for each $p$. For $p=2,3,5,7$ these sequences
appear in <a href="http://oeis.org/">OEIS</a> as sequences A000679, A090091, A090130, A090140.

A supplementary question is:
Does knowing if the classification is or is not smooth have any bearing on the complexity of these sequences?