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The functionality is available, because "the kernel of a matrix over Z/n" is 'the kernel of a morphism between finitely generated abelian groups" or (equivalently) "the kernel of a morphism between finitely generated Z-modules". To interpret the language you'll have to read some algebra texts that go beyond first year material, but the computations in the end all boil down to computing Hermite Normal Forms and possibly Smith Normal forms.

As an example, if you want to compute the (row) kernel of [[2,0],[0,3]] over Z/6Z you can proceed in the following way.

sage: Z2=ZZ^2
sage: M=Z2/(6*Z2)
sage: A=matrix([[2,0],[0,3]])
sage: phi=M.hom([M(a) for a in A])
sage: phi
Morphism from module over Integer Ring with invariants (6, 6) to module with invariants (6, 6) that sends the generators to [(0, 2), (3, 0)]
sage: phi.kernel()
Finitely generated module V/W over Integer Ring with invariants (6)
sage: [M(b) for b in phi.kernel()]
[(0, 0), (3, 4), (0, 2), (3, 0), (0, 4), (3, 2)]
sage: [M(b) for b in phi.kernel().gens()]
[(3, 4)]

This codes shows that the kernel itself looks like Z/6Z, what its 6 elements are, and that they are the Z/6Z multiples of the vector (3,4).