When n=2q is twice an odd prime power q, I think G = AGL(1,q) wr Sym(2) (amongst others) has the largest minimum orbit length on 2-sets. If G acts on Z = X ∪ Y, each of size q, then G has two orbits on Z(2), { {a,b} : a,b in X } ∪ { {a,b} : a,b in Y } of size 2⋅Binomial(q,2), and { {a,b} : a in X, b in Y } of size q2. When q is even, those are still the orbits, but of course AGL(1,2q) has only a single orbit.
When n=3q is three times a prime power q coprime to 3, then again AGL(1,q) wr Sym(3) looks reasonable, with only two orbits, one of size 3⋅Binomial(q,2), one of size 3⋅q2. At any rate, if it is not maximal it has a fairly large minimal orbit.
Obviously you can't keep taking Sym(k) for k=2,3,4,... forever, but I think actually it suffices to use the regular permutation representation Cyclic(k) of the cyclic group of order k. The largest orbit often splits into more than one orbit, but they are all so large it appears not to matter.
In other words, you might try AGL(1,q) wr Cyclic(k) for fairly general k and q, when n=k⋅q. I think the minimum orbit size will always be the silly disjoint union one of size k⋅Binomial(q,2), which is pretty large as long as you hold k constant. You can probably replace AGL(1,q) with your paper's Γ(q,d) without too much trouble.
I am under the impression that in asymptotic group theory, it is often quite hard to find exact sharp bounds, so just having "large" examples like these might be sufficient, at least for numbers with a very large prime power factor (so that k is kept small).
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