  We study the problem of transmission of information over classical and
classical-quantum channels in the one-shot regime where the underlying codes
are constrained to be group codes. In the achievability part, we introduce a
new input probability distribution that incorporates the encoding homomorphism
and the underlying channel law. Using a random coding argument, we characterize
the performance of group codes in terms of hypothesis testing relative-entropic
quantities. In the converse part, we establish bounds by leveraging a
hypothesis testing-based approach. Furthermore, we apply the one-shot result to
the asymptotic stationary memoryless setting, and establish a single-letter
lower bound on group capacities for both classes of channels. Moreover, we
derive a matching upper bound on the asymptotic group capacity.
