Three-point orbits of 2-transitive groups and their combinatorial applications

Let $G$ be a finite $2$-transitive permutation group. We determine the orbits of $G$ on the unordered triples of its point set, together with the three-point stabilizer and its image inside $S_3$, for every affine and almost simple family in the finite $2$-transitive classification. These data classify the $G$-invariant triple systems and the simple $2$-\mbox{$(v,3,\lambda )$} designs admitting $G$, and yield in a uniform way the standard objects attached to them: the incidence and block graphs (with arc-transitivity and strong regularity in the Steiner case), the constant-weight and design codes, and the Schurian coherent configurations on triples.