  In view of a potential interpretation of the role of the Mathieu group M_24
in the context of strings compactified on K3 surfaces, we develop techniques to
combine groups of symmetries from different K3 surfaces to larger 'overarching'
symmetry groups.
  We construct a bijection between the full integral homology lattice of K3 and
the Niemeier lattice of type (A_1)^24, which is simultaneously compatible with
the finite symplectic automorphism groups of all Kummer surfaces lying on an
appropriate path in moduli space connecting the square and the tetrahedral
Kummer surfaces. The Niemeier lattice serves to express all these symplectic
automorphisms as elements of the Mathieu group M_24, generating the
'overarching finite symmetry group' (Z_2)^4:A_7 of Kummer surfaces. This group
has order 40320, thus surpassing the size of the largest finite symplectic
automorphism group of a K3 surface by orders of magnitude. For every Kummer
surface this group contains the group of symplectic automorphisms leaving the
Kaehler class invariant which is induced from the underlying torus. Our results
are in line with the existence proofs of Mukai and Kondo, that finite groups of
symplectic automorphisms of K3 are subgroups of one of eleven subgroups of
M_23, and we extend their techniques of lattice embeddings for all Kummer
surfaces with Kaehler class induced from the underlying torus.

High Energy Physics - Theory;Algebraic Geometry;Group Theory