Index modulation (IM) significantly enhances the spectral efficiency of fluid antennas (FAs) enabled multiple-input multiple-output (MIMO) systems, which is named FA-IM. However, due to the dense distribution of ports on the FA, the wireless channel exhibits a high spatial correlation, leading to severe performance degradation in the existing FA-IM-assisted MIMO systems. To tackle this issue, this paper proposes a novel fluid antenna grouping index modulation (FA-GIM) scheme to mitigate the high correlation between the activated ports. Specifically, considering the characteristics of the FA two-dimensional (2D) surface structure and the spatially correlated channel model in FA-assisted MIMO systems, a block grouping method is adopted, where adjacent ports are assigned to the same group. Consequently, different groups independently perform port index selection and constellation symbol mapping, with only one port being activated within each group during each transmission interval. Then, a closed-form average bit error probability (ABEP) upper bound for the proposed scheme is derived. Numerical results show that, compared to state-of-the-art schemes, the proposed FA-GIM scheme consistently achieves significant bit error rate (BER) performance gains under various conditions. The proposed scheme is both efficient and robust, enhancing the performance of FA-assisted MIMO systems.