  We consider codes of length $m$ over an alphabet of size $q$ as subsets of
the vertex set of the Hamming graph $\Gamma=H(m,q)$. A code for which there
exists an automorphism group $X\leq Aut(\Gamma)$ that acts transitively on the
code and on its set of neighbours is said to be neighbour transitive, and were
introduced by the authors as a group theoretic analogue to the assumption that
single errors are equally likely over a noisy channel. Examples of neighbour
transitive codes include the Hamming codes, various Golay codes, certain
Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and
frequency permutation arrays, which have connections with powerline
communication, and also completely transitive codes, a subfamily of completely
regular codes, which themselves have attracted a lot of interest. It is known
that for any neighbour transitive code with minimum distance at least 3 there
exists a subgroup of $X$ that has a $2$-transitive action on the alphabet over
which the code is defined. Therefore, by Burnside's theorem, this action is of
almost simple or affine type. If the action is of almost simple type, we say
the code is alphabet almost simple neighbour transitive. In this paper we
characterise a family of neighbour transitive codes, in particular, the
alphabet almost simple neighbour transitive codes with minimum distance at
least $3$, and for which the group $X$ has a non-trivial intersection with the
base group of $Aut(\Gamma)$. If $C$ is such a code, we show that, up to
equivalence, there exists a subcode $\Delta$ that can be completely described,
and that either $C=\Delta$, or $\Delta$ is a neighbour transitive frequency
permutation array and $C$ is the disjoint union of $X$-translates of $\Delta$.
  We also prove that any finite group can be identified in a natural way with a
neighbour transitive code.
