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Spatially extended halos of H I Ly$\alpha$ emission are now ubiquitously found around high-redshift star-forming galaxies. But our understanding of the nature and powering mechanisms of these halos is still hampered by the complex radiative transfer effects of the Ly$\alpha$ line and limited angular resolution. In this paper, we present resolved Multi Unit Spectroscopic Explorer (MUSE) observations of SGAS J122651.3+215220, a strongly-lensed pair of $L^{*}$ galaxies at $z=2.92$ embedded in a Ly$\alpha$ halo of $L_{Ly\alpha}=(6.2\pm1.3)\times10^{42}$ erg s$^{-1}$. Globally, the system shows a line profile that is markedly asymmetric and redshifted, but its width and peak shift vary significantly across the halo. By fitting the spatially binned Ly$\alpha$ spectra with a collection of radiative transfer galactic wind models, we infer a mean outflow expansion velocity of $\approx 211$ km s$^{-1}$, with higher values preferentially found on both sides of the system's major axis. The velocity of the outflow is validated with the blueshift of low-ionization metal absorption lines in the spectra of the central galaxies. We also identify a faint ($M_{1500} \approx -16.7$) companion detected in both Ly$\alpha$ and the continuum, whose properties are in agreement with a predicted population of satellite galaxies that contribute to the extended Ly$\alpha$ emission. Finally, we briefly discuss the impact of the interaction between the central galaxies on the properties of the halo and the possibility of in situ fluorescent Ly$\alpha$ production., Comment: Published in ApJ. 26 pages, 14 figures