I would like a method to efficiently generate a random finite group of a given order $n$. If there are $g(n)$ non-isomorphic groups of order $n$, ideally each group would occur with probability $1/g(n)$. So if $n=64$, each of the $267$ groups would be generated with equal probability. ($g(n)$ is A000001 in OEIS.) Groups of order $n=2^k$ would be of special interest.
This is far from my expertise, and my searches must be using the wrong terminology, because I have not found such methods. I'd appreciate pointers—Thanks!
Addendum. The comments indicate that this appears to be an open problem, with little chance of resolution in the near future. Now so tagged.