  We provide a method, based on Nikulin's lattice gluing techniques, which
identifies the symplectic automorphisms of Kummer surfaces as permutation
groups on 24 elements preserving the Golay code. In other words, we explicitly
realise these symplectic automorphism groups as subgroups of the Mathieu group
M_24. The example of the tetrahedral Kummer surface is treated in detail,
confirming the existence proofs of Mukai and Kondo, that its group of
symplectic automorphisms is a subgroup of one of eleven subgroups of the
sporadic group known as Mathieu group M_23. Kondo's lattice construction, which
uses a different gluing technique from the one advocated here to rederive
Mukai's results, is reviewed, and a slight generalisation is used to check the
consistency of our results. The framework presented here provides a line of
attack to unravel the role of the sporadic Mathieu group Mathieu M_24, of which
M_23 is a subgroup of index 24, when searching for symmetries beyond the
classical symplectic automorphisms in the context of strings compactified on a
K3 surface.
