As post hoc explanation methods are increasingly being leveraged to explain complex models in high-stakes settings, it becomes critical to ensure that the quality of the resulting explanations is consistently high across various population subgroups including the minority groups. For instance, it should not be the case that explanations associated with instances belonging to a particular gender subgroup (e.g., female) are less accurate than those associated with other genders. However, there is little to no research that assesses if there exist such group-based disparities in the quality of the explanations output by state-of-the-art explanation methods. In this work, we address the aforementioned gaps by initiating the study of identifying group-based disparities in explanation quality. To this end, we first outline the key properties which constitute explanation quality and where disparities can be particularly problematic. We then leverage these properties to propose a novel evaluation framework which can quantitatively measure disparities in the quality of explanations output by state-of-the-art methods. Using this framework, we carry out a rigorous empirical analysis to understand if and when group-based disparities in explanation quality arise. Our results indicate that such disparities are more likely to occur when the models being explained are complex and highly non-linear. In addition, we also observe that certain post hoc explanation methods (e.g., Integrated Gradients, SHAP) are more likely to exhibit the aforementioned disparities. To the best of our knowledge, this work is the first to highlight and study the problem of group-based disparities in explanation quality. In doing so, our work sheds light on previously unexplored ways in which explanation methods may introduce unfairness in real world decision making.