  Binary Parseval frames share many structural properties with real and complex
ones. On the other hand, there are subtle differences, for example that the
Gramian of a binary Parseval frame is characterized as a symmetric idempotent
whose range contains at least one odd vector. Here, we study binary Parseval
frames obtained from the orbit of a vector under a group representation, in
short, binary Parseval group frames. In this case, the Gramian of the frame is
in the algebra generated by the right regular representation. We identify
equivalence classes of such Parseval frames with binary functions on the group
that satisfy a convolution identity. This allows us to find structural
constraints for such frames. We use these constraints to catalogue equivalence
classes of binary Parseval frames obtained from group representations. As an
application, we study the performance of binary Parseval frames generated with
abelian groups for purposes of error correction. We show that $Z_p^q$ is always
preferable to $Z_{p^q}$ when searching for best performing codes associated
with binary Parseval group frames.
