Entry-faithful 2-neighbour transitive codes

We consider a code to be a subset of the vertex set of a Hamming graph. The set of $$s$$s-neighbours of a code is the set of vertices, not in the code, at distance $$s$$s from some codeword, but not distance less than $$s$$s from any codeword. A $$2$$2-neighbour transitive code is a code which admits a group $$X$$X of automorphisms which is transitive on the $$s$$s-neighbours, for $$s=1,2$$s=1,2, and transitive on the code itself. We give a classification of $$2$$2-neighbour transitive codes, with minimum distance $$\delta \geqslant 5$$δ⩾5, for which $$X$$X acts faithfully on the set of entries of the Hamming graph.