We present a combined X-ray, optical, and radio analysis of the galaxy group IC 1860 using the currently available Chandra and XMM data, literature multi-object spectroscopy data and GMRT data. The Chandra and XMM imaging and spectroscopy reveal two surface brightness discontinuities at 45 and 76 kpc shown to be consistent with a pair of cold fronts. These features are interpreted as due to sloshing of the central gas induced by an off-axis minor merger with a perturber. This scenario is further supported by the presence of a peculiar velocity of the central galaxy IC 1860 and the identification of a possible perturber in the optically disturbed spiral galaxy IC 1859. The identification of the perturber is consistent with the comparison with numerical simulations of sloshing. The GMRT observation at 325 MHz shows faint, extended radio emission contained within the inner cold front, as seen in some galaxy clusters hosting diffuse radio mini-halos. However, unlike mini-halos, no particle reacceleration is needed to explain the extended radio emission, which is consistent with aged radio plasma redistributed by the sloshing. There is strong analogy of the X-ray and optical phenomenology of the IC 1860 group with two other groups, NGC 5044 and NGC 5846, showing cold fronts. The evidence presented in this paper is among the strongest supporting the currently favored model of cold-front formation in relaxed objects and establishes the group scale as a chief environment to study this phenomenon.