Suppose that \(\cal{D}\) is a 2-(v, k, λ) design with (v − 1, k − 1) = 2 and that G is a flag-transitive group of automorphisms of \(\cal{D}\) . It is shown in this paper that either G is 2-transitive on points or G is a subgroup of the group AΓL(1, v) of 1-dimensional semilinear affine transformations, so that \(\cal{D}\) has v = pd points (p is a prime). Further, we classify such type of designs admitting an almost simple automorphism group G with socle a Suzuki group Sz(q), by considering the classification under the (even weaker) assumption that k is odd. As its application, we construct two new families of flag-transitive 2-designs with socle Sz(q).
