The galaxy cluster Abell 496 has been extensively studied in the past for the clear sloshing motion of its hot intracluster medium (ICM) on large scales, but the interplay between the central radio galaxy and the surrounding cluster atmosphere is mostly unexplored. We present a dedicated radio, X-ray, and optical study of Abell 496 with the aim being to investigate this connection. We use deep radio images obtained with the Giant Metrewave Radio Telescope (GMRT) at 150, 330, and 617 MHz, the Very Large Array (VLA) at 1.4 and 4.8 GHz, and the VLA Low Band Ionosphere and Transient Experiment (VLITE) at 340 MHz, with angular resolutions ranging from $0.^ prime 5$ to $25^ prime $. Additionally, we use archival Chandra and Very Large Telescope (VLT) MUSE observations. The radio images reveal three distinct periods of jet activity: an ongoing episode on subkiloparsec scales with an inverted radio spectrum; an older episode that produced lobes on scales of $ kpc, which now have a steep spectral index ($ and an black even older episode that produced lobes on scales of $ kpc with an ultrasteep spectrum ($ Archival Chandra X-ray observations show that the older and oldest episodes excavated two generations of cavities in the hot gas of the cluster. The outermost X-ray cavity has a clear mushroom-head shape, likely caused by its buoyant rise in the cluster's potential. Cooling of the hot gas is ongoing in the innermost 20 kpc, where warm, Halpha -bright filaments are visible in VLT-MUSE data. The Halpha -filaments are stretched toward the mushroom-head cavity, which may have stimulated ICM cooling in its wake. We conclude by discussing our nondetection of a radio mini-halo in this vigorously sloshing but low-mass galaxy cluster.