The richest 1% on the planet own almost half of the world’s wealth, according to a study by Oxfam International published in 2021. This marked inequality in the distribution of wealth is of major concern to our society and economy. Several measures are known to quantify inequality in the distribution of a variable of interest, the Gini index being the most popular. However, the literature suggests that this inequality measure is unable to discriminate between different tail behaviours. For instance, heavy-tailed and short-tailed distributions may have the same Gini index. In this work, we rigorously state that the Gini index does not discriminate between maximum domains of attraction. To overcome this drawback, an "extreme" Gini index is introduced, based on the excesses over a high threshold. We show that, when the threshold tends towards the endpoint, this extreme Gini index is able to discriminate between different distribution tails.