We propose a robust approach to recovering the jointly-sparse signals in the presence of outliers. We formulate the recovering task as a minimization problem involving three terms: (i) the minimax concave (MC) loss function, (ii) the MC penalty function, and (iii) the squared Frobenius norm. The MC-based loss and penalty functions enhance robustness and group sparsity, respectively, while the squared Frobenius norm induces the convexity. The problem is solved, via reformulation, by the primal-dual splitting method, for which the convergence condition is derived. Numerical examples show that the proposed approach enjoys remarkable outlier robustness.