A 2-(n,4,λ) design (Ω,B) is said to be supersimple if distinct lines intersect in at most two points. From such a design, one can construct a certain subset of Sym(Ω) called a "Conway groupoid". The construction generalizes Conway's construction of the groupoid M₁₃. % One would like to classify all of the Conway groupoids constructed using supersimple designs. In this paper we classify a particular subclass, consisting of those groupoids which satisfy two additional properties: Firstly the set of collinear point-triples forms a regular two-graph, and secondly the symmetric difference of two intersecting lines is again a line. % The proof uses Hall's work on $3$-transposition groups of symplectic type, and Seidel's work on graphs that satisfy the triangle property.