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 10481 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 OEIS 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?