In the literature on nonnegative matrix theory, real square invertible matrices with the property that their inverses are M-matrices have been extensively studied. Real square (group invertible) singular M-matrices whose group inverses are M-matrices also have received a lot of attention. The endeavour of the present work is to study the more general class of all group invertible matrices (not necessarily singular M-matrices) whose group inverses are M-matrices. Among other things, a characterization is proved. Many results, motivated by the corresponding ones for inverse M-matrices, are obtained. Let A be any 2×2 matrix or a 3×3 symmetric, tridiagonal matrix. A complete characterization for A to posses the property that its group inverse is an M-matrix, is presented. © 2019 Elsevier Inc.