  This article continues a prior investigation of the authors with the goal of
extending characterization results of convolutional tight frames from the
context of cyclic groups to general finite abelian groups. The collections
studied are formed by translating a number of \emph{generators} by elements of
a fixed subgroup and it is shown, under certain norm conditions, that tight
frames with this structure are characterized as local minimizers of the frame
potential. Natural analogs to the downsampling and upsampling operators of
finite cyclic groups are studied for arbitrary subgroups of finite abelian
groups. Directions of further study are also proposed.
