Non-split sharply 2-transitive groups of odd positive characteristic

It is well-known that every sharply 2-transitive group of characteristic 3 splits. Here we construct the first examples of non-split sharply 2-transitive groups in odd positive characteristic p, for sufficiently large primes p. Furthermore, we show that any group without 2-torsion can be embedded into a non-split sharply 2-transitive group of characteristic p for all sufficiently large primes p, yielding 2^ℵ_0 many pairwise non-isomorphic countable non-split sharply 2-transitive groups in any sufficiently large characteristic.