If I understood Joel David Hamkins's explanation correctly, then the problem of classifying finite ($p$-)groups is smooth for trivial reasons. Attach to each group the following datum: randomly number the elements of a group $G$ from 1 to $|G|$ and consider the set $S_G$ of multiplication tables of $G$ with respect to all possible permutations of the elements, so $(|G|)!$ multiplication tables in total. Two groups $G$ and $H$ are isomorphic if and only if the sets $S_G$ and $S_H$ are equal.

If I understood Joel David Hamkins's explanation correctly, then the problem of classifying finite ($p$-)groups is smooth for silly reasons. Attach to each group the following datum: randomly number the elements of a group $G$ from 1 to $|G|$ and consider the set $S_G$ of multiplication tables of $G$ with respect to all possible permutations of the elements, so $(|G|)!$ multiplication tables in total. Two groups $G$ and $H$ are isomorphic if and only if the sets $S_G$ and $S_H$ are equal.