Automorphism groups of certain simple 2-(q,3,λ) designs constructed from finite fields

Let F be a finite field of characteristic not 2, and S F a subset with three elements. Consider the collection S={S•a+b a,b F, a≠0}.Then (F,S) is a simple 2-design and the parameter λ of (F,S) is 1, 2, 3 or 6. We find in this paper the full automorphism group of (F,S). Namely, if we put U={r {0,1,r} S} and K the subfield of F generated by U, then the automorphisms of (F,S) are the maps of the form x g(α(x))+b, x F, where b F, α : F→F is a field automorphism fixing U, and g is a linear transformation of F considered as a vector space over K.