We study the one-shot channel coding problem over classical and classical-quantum channels, where the underlying codes are constrained to be group codes. In the achievability part, we introduce a new distribution that incorporates the encoding homomorphism and the underlying channel law. Using a random coding argument, we characterize the performance in terms of hypothesis testing relative-entropies. In the converse part, we establish bounds by leveraging a hypothesis testing-based approach. Further we apply the one-shot results to the asymptotic use case and establish the group capacities for both channels.