Time-varying group interactions constitute the building blocks of many complex systems. The framework of temporal hypergraphs makes it possible to represent them by taking into account the higher-order and temporal nature of the interactions. However, the corresponding datasets are often incomplete and/or limited in size and duration, and surrogate time-varying hypergraphs able to reproduce their statistical features constitute interesting substitutions, especially to understand how dynamical processes unfold on group interactions. Here, we present a new temporal hypergraph model, the Emerging Activity Temporal Hypergraph (EATH), which can be fed by parameters measured in a dataset and create synthetic datasets with similar properties. In the model, each node has an independent underlying activity dynamic and the overall system activity emerges from the nodes dynamics, with temporal group interactions resulting from both the activity of the nodes and memory mechanisms. We first show that the EATH model can generate surrogate hypergraphs of several empirical datasets of face-to-face interactions, mimicking temporal and topological properties at the node and hyperedge level. We also showcase the possibility to use the resulting synthetic data in simulations of higher-order contagion dynamics, comparing the outcome of such process on original and surrogate datasets. Finally, we illustrate the flexibility of the model, which can generate synthetic hypergraphs with tunable properties: as an example, we generate "hybrid" temporal hypergraphs, which mix properties of different empirical datasets. Our work opens several perspectives, from the generation of synthetic realistic hypergraphs describing contexts where data collection is difficult to a deeper understanding of dynamical processes on temporal hypergraphs.