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A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is$2\it{\text{-neighbour-transitive}}$ if ${\rm Aut}(C)$ acts transitively oneach of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distancepartition of $V\varGamma$ with respect to $C$. Previous classifications offamilies of $2$-neighbour-transitive codes leave only those with an affineaction on the alphabet to be investigated. Here, $2$-neighbour-transitive codeswith minimum distance at least $5$ and that contain "small" subcodes as blocksof imprimitivity are classified. When considering codes with minimum distanceat least $5$, completely transitive codes are a proper subclass of$2$-neighbour-transitive codes. Thus, as a corollary of the main result,completely transitive codes satisfying the above conditions are alsoclassified.

$2$-Neighbour-Transitive Codes with Small Blocks of Imprimitivity