  A submodule of a $\mathbb{Z}$-module determines a coloring of the module
where each coset of the submodule is associated to a unique color. Given a
submodule coloring of a $\mathbb{Z}$-module, the group formed by the symmetries
of the module that induces a permutation of colors is referred to as the color
group of the coloring. In this contribution, a method to solve for the color
groups of colorings of $N$-planar modules where $N=4$ and $N=6$ are given.
Examples of colorings of rectangular lattices and of the vertices of the
Ammann-Beenker tiling are given to exhibit how these methods may be extended to
the general case.
