Models of opinion dynamics traditionally consider the evolution of opinions defined on nodes of a network whose links represent pairwise interactions between the nodes. However, in many situations groups of nodes might be considered, for all practical purposes, to also possess an opinion. To study the effect of groups on opinion dynamics, in this paper we study an opinion model where both nodes and groups of nodes have opinions and affect each other via pairwise interactions and group memberships. We find that the addition of group opinions can result, in some parameter regimes, in oscillatory and excitable opinion dynamics. In the excitable opinion dynamics regime, finite-size effects create large but short-lived opinion swings. In the oscillatory regime, the average opinion oscillates periodically. We develop a mean-field description of our model based on a hypergraph formulation and find good agreement with simulations of the model. Using the mean-field description and numerical simulations, we show that oscillatory dynamics only occur when the structure of pairwise interactions is different from the structure of group memberships. Our results are a clear example of how higher-order structures, such as groups of nodes endowed with opinions, can have important effects on the collective dynamics of opinions.