Independent component analysis (ICA) has been successfully applied for single-subject functional Magnetic Resonance Imaging (fMRI) analysis. The extension of ICA for group inferences remains an active research topic. We consider two popular group ICA methods for multi-subject fMRI data: GIFT and tensor PICA. Both models assume statistical independence in the spatial source signals in the brain. The two methods differ in terms of the assumed structure of the group spatio-temporal process. Their estimation and inference procedures are also different and tailored specifically for the assumed model structure. To understand the relative effectiveness of the two methods under various scenarios, we generate multi-subject fMRI data to compare the performance the two methods under varying types of between-subject variability. The simulation results show that both methods have high accuracy when the data were generated from the trilinear model[1]. When the underlying model deviates from the trilinear model, GIFT tends to be more robust than the tensor PICA. We reveal the connection between the two methods by introducing a class of group ICA models. We show that GIFT (Calhoun et al., 2001. Hum Brain Mapp, 14: 140-151) and tensor PICA (Beckmann and Smith, 2005. NeuroImage, 25: 294-311) are special cases in this class of models when assuming different model structures.