ABSTRACT This is the third paper of a series where we study the stellar population gradients (SP; ages, metallicities, α-element abundance ratios, and stellar initial mass functions) of early-type galaxies (ETGs) at $z$ ≤ 0.08 from the Mapping Nearby Galaxies at APO Data Release 15 (MaNGA-DR15) survey. In this work, we focus on the S0 population and quantify how the SP varies across the population as well as with galactocentric distance. We do this by measuring Lick indices and comparing them to SP synthesis models. This requires spectra with high signal-to-noise ratio which we achieve by stacking in bins of luminosity (Lr) and central velocity dispersion (σ0). We find that: (1) there is a bimodality in the S0 population: S0s more massive than $3\times 10^{10}\, \mathrm{M}_⊙$ show stronger velocity dispersion and age gradients (age and σr decrease outwards) but little or no metallicity gradient, while the less massive ones present relatively flat age and velocity dispersion profiles, but a significant metallicity gradient (i.e. [M/H] decreases outwards). Above $2\times 10^{11}\, \mathrm{M}_⊙$, the number of S0s drops sharply. These two mass scales are also where global scaling relations of ETGs change slope. (2) S0s have steeper velocity dispersion profiles than fast-rotating elliptical galaxies (E-FRs) of the same luminosity and velocity dispersion. The kinematic profiles and SP gradients of E-FRs are both more similar to those of slow-rotating ellipticals (E-SRs) than to S0s, suggesting that E-FRs are not simply S0s viewed face-on. (3) At fixed σ0, more luminous S0s and E-FRs are younger, more metal rich and less α-enhanced. Evidently for these galaxies, the usual statement that ‘massive galaxies are older’ is not true if σ0 is held fixed.