How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?

I just came across this question and, even though I'm a bit late, I thought you might be interested in this reference:

Even, S.; Goldreich, O., The minimum-length generator sequence problem is NP-hard, J. Algorithms 2, 311-313 (1981). ZBL0467.68046.

This asserts that the calculation of the diameter of a Cayley graph of a permutation group is NP-Hard. I don't have access to the article myself but, from what I've read elsewhere, I believe this statement remains true even if you restrict to elementary-abelian 2-groups. (Which seems astonishing to me!)

This seems quite difficult, for an example of results (and an indication of the difficulty) see Ganesan's 2011 paper.