Sharply 2-transitive permutation groups are studied in this work. The notions of transversal in a group [3] and Sabinin’s semidirect product [5] are used. All elements of order 2 from a sharply 2-transitive permutation group G form a loop transversal T in G to H0 = St0(G). For an arbitrary element ti ∈ T (ti 6= id) its centralizer Ci = CG(ti) is a group transversal in G to H0 = St0(G). The group G may be represented as a semidirect product: G = T ⋋ H0 = Ci ⋋ H0. A construction of the group G as an external semidirect product of two suitable groups is described. It gives us a potential example of an infinite sharply 2-transitive permutation group G with a non-abelian normal subgroup T , which consists of fixed-point-free permutations and the identity permutation.