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](https://oeis.org/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.