I wrote:
Up to order 255, the only group that exceeds 1/6 is SmallGroup(48,50), or
<a,b,c : a^3 = b^2 = c^2 = (a b)^3 = (a c)^3 = (b c)^2 = (a b a^-1 c)^2 = 1> .
It has a probability of 5/24 that a,b,ab are square,square,nonsquare.
I looked at your classification of groups of order 16 and verified your translation of the smaller anonymous groups I found. In hopes of identifying this group, I found its derived subgroup 2^4 . Of course G/G' is C3. But the direct product C3 x 2^4 is not G. That seems similar to the case for A4: A4/C3 = 2^2 but 2^2x3 is not A4. Somehow I thought that was the way to decompose and reconstitute groups, but I guess I'm mistaken. As for identifying the group, I have checked that its order spectrum [1,15,32,0,...,0] is unique. Its center is 1. Dan