This study investigates mask-based beamformers (BFs), which estimate filters for target sound extraction (TSE) using time-frequency masks. Although multiple mask-based BFs have been proposed, no consensus has been reached on which one offers the best target-extraction performance. Previously, we found that maximum signal-to-noise ratio and minimum mean square error (MSE) BFs can achieve the same extraction performance as the theoretical upper-bound performance, with each BF containing a different optimal mask. However, two issues remained unsolved: only two BFs were covered, excluding the minimum variance distortionless response BF, and ideal scaling (IS) was employed to ideally adjust the output scale, which is not applicable to realistic scenarios. To address these issues, this study proposes a unified framework for mask-based BFs comprising two processes: filter estimation that can cover all possible BFs and scaling applicable to realistic scenarios by employing a mask to generate a scaling reference. Based on the operators and covariance matrices used in BF formulas, all possible BFs can be classified into 12 variations, including two new ones. Optimal masks for both processes are obtained by minimizing the MSE between the target and BF output. The experimental results using the CHiME-4 dataset suggested that 1) all 12 variations can achieve the theoretical upper-bound performance, and 2) mask-based scaling can behave like IS, even when constraining the temporal mean of a non-negative mask to one. These results can be explained by considering the practical parameter count of the masks. These findings contribute to 1) designing a TSE system, 2) improving scaling accuracy through mask-based scaling, and 3) estimating the extraction performance of a BF.