  Given a group G, the model \mathcal{G}(G,p) denotes the probability space of
all Cayley graphs of G where each element of the generating set is chosen
independently at random with probability p.
  Given a family of groups (G_k) and a c \in \mathbb{R}_+ we say that c is the
threshold for diameter 2 for (G_k) if for any \varepsilon > 0 with high
probability \Gamma \in \mathcal{G}(G_k,p) has diameter greater than 2 if p
\leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}} and diameter at most 2 if p
\geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}. In [5] we proved that if c is a
threshold for diameter 2 for a family of groups (G_k) then c \in [1/4,2] and
provided two families of groups with thresholds 1/4 and 2 respectively.
  In this paper we study the question of whether every c \in [1/4,2] is the
threshold for diameter 2 for some family of groups. Rather surprisingly it
turns out that the answer to this question is negative. We show that every c
\in [1/4,4/3] is a threshold but a c \in (4/3,2] is a threshold if and only if
it is of the form 4n/(3n-1) for some positive integer n.
