This paper studies a new kind of generalized beta distribution that is different from the GB1 and GB2 of McDonald (1984). This new four-parameter statistical distribution, the extended k-generalized distribution of the second kind, abbreviated EkG2, is derived as one of two kinds of generalizations from the k-generalized distribution of Clementi et al. (2007). By empirical comparison with the GB2 using the LIS income/consumption data, the EkG2 is found to be an overall better fit in terms of both frequency-based (FB) evaluation criteria, such as the likelihood, and money-amount-based (MAB) evaluation criteria, such as the accuracy of the estimated Lorentz curve. The EkG2 also overall outperforms the double Pareto-lognormal distribution (dPLN) of Reed (2003) in terms of FB criteria. Although not necessarily superior to the dPLN in terms of MAB criteria, the EkG2 is judged to be an overall better fit to the empirical distributions relative to the dPLN by a combined evaluation using both FB and MAB criteria. This paper also discusses similarities and differences in characteristics between the EkG2 and GB2, including the shapes of the distributions.