Motivated by a recent paper, in which the authors studied Re-nnd $\{1,3\}$-inverse, $\{1,4\}$-inverse and $\{1,3,4\}$-inverse of a square matrix, in this paper, we establish some equivalent conditions for the existence of Re-nnd $\{1,2,3\}$-inverse, $\{1,2,4\}$-inverse and $\{1,3,4\}$-inverse. Furthermore, some expressions of these generalized inverses are presented.