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Using Multi-Unit Spectroscopic Explorer (MUSE) spectroscopy, we analyse the stellar kinematics of 18 brightest group early-type (BGEs) galaxies, selected from the Complete Local-Volume Groups Sample (CLoGS). We analyse the kinematic maps for distinct features, and measure specific stellar angular momentum within one effective radius ($λ_{e}$). We classify the BGEs as fast (10/18) or slow (8/18) rotators, suggesting at least two different evolution paths. We quantify the anti-correlation between higher-order kinematic moment $h_{3}$ and V/$σ$ (using the $ξ_{3}$ parameter), and the kinematic misalignment angle between the photometric and kinematic position angles (using the $Ψ$ parameter), and note clear differences between these parameter distributions of the fast and slow rotating BGEs. We find that all 10 of our fast rotators are aligned between the morphological and kinematical axis, consistent with an oblate galaxy shape, whereas the slow rotators are spread over all three classes: oblate (1/8), triaxial (4/8), and prolate (3/8). We place the results into context using known radio properties, X-ray properties, and observations of molecular gas. We find consistent merger histories inferred from observations for the fast-rotating BGEs, indicating that they experienced gas-rich mergers or interactions, and these are very likely the origin of the cold gas. Observational evidence for the slow rotators are consistent with gas-poor mergers. For the slow rotators with cold gas, all evidence point to cold gas cooling from the intragroup medium.