  We construct super vertex operator algebras which lead to modules for
moonshine relations connecting the four smaller sporadic simple Mathieu groups
with distinguished mock modular forms. Starting with an orbifold of a free
fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its
unique non-trivial 24-dimensional representation commutes with a certain N=4
superconformal algebra. Similarly, any subgroup of Co_0 that fixes a
2-dimensional subspace of the 24-dimensional representation commutes with a
certain N=2 superconformal algebra. Through the decomposition of the
corresponding twined partition functions into characters of the N=4 (resp. N=2)
superconformal algebra, we arrive at mock modular forms which coincide with the
graded characters of an infinite-dimensional Z-graded module for the
corresponding group. The Mathieu groups are singled out amongst various other
possibilities by the moonshine property: requiring the corresponding weak
Jacobi forms to have certain asymptotic behaviour near cusps. Our constructions
constitute the first examples of explicitly realized modules underlying
moonshine phenomena relating mock modular forms to sporadic simple groups.
Modules for other groups, including the sporadic groups of McLaughlin and
Higman--Sims, are also discussed.
