Objective. Demographic groups are often represented at different rates in medical datasets. These differences can create bias in machine learning algorithms, with higher levels of performance for better-represented groups. One promising solution to this problem is to generate synthetic data to mitigate potential adverse effects of non-representative data sets. Methods. We build on recent advances in LLM-based synthetic data generation to create a pipeline where the synthetic data is generated separately for each demographic group. We conduct our study using MIMIC-IV and Framingham "Offspring and OMNI-1 Cohorts" datasets. We prompt GPT4-Turbo to create group-specific data, providing training examples and the dataset context. An exploratory analysis is conducted to ascertain the quality of the generated data. We then evaluate the utility of the synthetic data for augmentation of a training dataset in a downstream machine learning task, focusing specifically on model performance metrics across groups. Results. The performance of GPT4-Turbo augmentation is generally superior but not always. In the majority of experiments our method outperforms standard modeling baselines, however, prompting GPT-4-Turbo to produce data specific to a group provides little to no additional benefit over a prompt that does not specify the group. Conclusion. We developed a method for using LLMs out-of-the-box to synthesize group-specific data to address imbalances in demographic representation in medical datasets. As another "tool in the toolbox", this method can improve model fairness and thus health equity. More research is needed to understand the conditions under which LLM generated synthetic data is useful for non-representative medical data sets.