  We define Macaulay bases of modules, which are a common generalization of
Groebner bases and Macaulay $H$-bases to suitably graded modules over a
commutative graded $\mathbf{k}$-algebra, where the index sets of the two
gradings may differ. This includes Groebner bases of modules as a special case,
in contrast to previous work on Macaulay bases of modules. We show that the
standard results on Groebner bases and Macaulay $H$-bases generalize in fields
of arbitrary characteristic to Macaulay bases, including the reduction
algorithm and Buchberger's criterion and algorithm. A key result is that
Macaulay bases, in contrast to Groebner bases, respect symmetries when there is
a group $G$ acting homogeneously on a graded module, in which case the
reduction algorithm is $G$-equivariant and the $\mathbf{k}$-span of a Macaulay
basis is $G$-invariant. We also show that some of the standard applications of
Groebner bases can be generalized to Macaulay bases, including elimination and
computation of syzygy modules, which require the generalization to modules that
was not present in previous work.
