We present absorption-line strength maps of a sample of 24 representative early-type spiral galaxies, mostly of type Sa, obtained as part of the SAURON (Spectrographic Areal Unit for Research on Optical Nebulae) survey of nearby galaxies using our custom-built integral-field spectrograph. Using high-quality spectra, spatially binned to a constant signal-to-noise ratio, we measure several key age, metallicity and abundance ratio sensitive indices from the Lick Observatory Image Dissector Scanner (Lick/IDS) system over a contiguous two-dimensional field including bulge and inner disc. We present maps of Hβ, Fe5015 and Mgb for each galaxy. We find that Sa galaxies on the average have slightly smaller Mgb and Fe5015 line strengths than ellipticals and S0s, and higher Hβ values, but with a much larger scatter. The absorption-line maps show that many galaxies contain some younger populations (<=1Gyr), distributed in small or large inner discs, or in circumnuclear star-forming rings. In many cases these young stars are formed in circumnuclear ministarbursts, which are dominating the light in the centres of some of the early-type spirals. These ministarburst cause a considerable scatter in index-index diagrams such as Mgb-Hβ and Mgb-Fe5015, more than is measured for early-type galaxies. We find that the central regions of Sa galaxies display a wide range in ages, even within the galaxies. We find that the central regions of early-type spirals are often dusty, with a good correlation between the presence of young central stellar populations and a significant amount of dust extinction. 50 per cent of the sample show velocity dispersion drops in their centres. All of the galaxies of our sample lie on or below the Mgb -σ relation for elliptical galaxies in the Coma cluster, and above the Hβ absorption line-σ relation for elliptical galaxies. If those relations are considered to be relations for the oldest local galaxies we see that our sample of spirals has a considerable scatter in age, with the largest scatter at the lowest σ. This is in disagreement with highly inclined samples, in which generally only old stellar populations are found in the central regions. The discrepancy between our sample and highly inclined samples, and the presence of so many stellar velocity dispersion dips, i.e. so-called σ drops, in these spiral galaxies with large bulges (type Sa) can be understood if the central regions of Sa galaxies contain at least two components: a thin, disc-like component, often containing recent star formation, and another, elliptical-like component, consisting of old stars and rotating more slowly, dominating the light above the plane. These components together form the photometrically defined bulge, in the same way as the thin and the thick disc co-exist in the solar neighbourhood. In this picture, consistent with the current literature, part of the bulge, the thicker component, formed a very long time ago. Later, stars continued to form in the central regions of the disc, rejuvenating in this way the bulge through dynamical processes. This picture is able to explain in a natural way the heterogeneous stellar populations and star formation characteristics that we are seeing in detailed observations of early-type spiral galaxies.