Inferring effective brain connectivity from neuroimaging data such as functional Magnetic Resonance Imaging (fMRI) has been attracting increasing interest due to its critical role in understanding brain functioning. Incorporating sparsity into connectivity modeling to make models more biologically realistic and performing group analysis to deal with inter-subject variability are still challenges associated with fMRI brain connectivity modeling. To address the above two crucial challenges, the attractive computational and theoretical properties of the least absolute shrinkage and selection operator (LASSO) in sparse linear regression provide a suitable starting point. We propose a group robust LASSO (grpRLASSO) model by combining advantages of the popular group-LASSO and our recently developed robust-LASSO. Here group analysis is formulated as a grouped variable selection procedure. Superior performance of the proposed grpRLASSO in terms of group selection and robustness is demonstrated by simulations with large noise variance. The grpRLASSO is also applied to a real fMRI data set for brain connectivity study in Parkinson's disease, resulting in biologically plausible networks.