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