Axelrod's model describes the dissemination of a set of cultural traits in a society constituted by individual agents. In a social context, nevertheless, individual choices toward a specific attitude are also at the basis of the formation of communities, groups and parties. The membership in a group changes completely the behavior of single agents who start acting according to a social identity. Groups act and interact among them as single entities, but still conserve an internal dynamics. We show that, under certain conditions of social dynamics, the introduction of group dynamics in a cultural dissemination process avoids the flattening of the culture into a single entity and preserves the multiplicity of cultural attitudes. We also consider diffusion processes on this dynamical background, showing the conditions under which information as well as innovation can spread through the population in a scenario where the groups' choices determine the social structure.