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 fluid antennas, the wireless channel exhibits a high spatial correlation, resulting in severe performance degradation in the existing FA-IM scheme. This paper proposes a novel fluid antenna grouping index modulation (FA-GIM) scheme to mitigate the spatial correlation of the FA-IM channel, further enhancing system performance. Based on the spatial correlation model of two-dimensional (2D) fluid antenna surfaces, this paper specifically adopts a block grouping method where adjacent ports are allocated to the same group. The numerical results demonstrate that the proposed scheme exhibits superior bit error rate (BER) performance compared to the state-of-the-art scheme, enhancing the robustness of FA-assisted MIMO systems.