We report on the star formation histories and extinction in the central kpc region of a sample of starburst galaxies that have similar far infrared (FIR), 10 micron and K-band luminosities as those of the archetype starburst M82. Our study is based on new optical spectra and previously published K-band photometric data, both sampling the same area around the nucleus. Model starburst spectra were synthesized as a combination of stellar populations of distinct ages formed over the Hubble time, and were fitted to the observed optical spectra and K-band flux. The model is able to reproduce simultaneously the equivalent widths of emission and absorption lines, the continuum fluxes between 3500-7000 Ang, the K-band and the FIR flux. We require a minimum of 3 populations -- (1) a young population of age < 8 Myr, with its corresponding nebular emission, (2) an intermediate-age population (age < 500 Myr), and (3) an old population that forms part of the underlying disk or/and bulge population. The contribution of the old population to the K-band luminosity depends on the birthrate parameter and remains above 60% in the majority of the sample galaxies. Even in the blue band, the intermediate age and old populations contribute more than 40% of the total flux in all the cases. A relatively high contribution from the old stars to the K-band nuclear flux is also apparent from the strength of the 4000 Ang break and the CaII K line. The extinction of the old population is found to be around half of that of the young population. The contribution to the continuum from the relatively old stars has the effect of diluting the emission equivalent widths below the values expected for young bursts. The mean dilution factors are found to be 5 and 3 for the Halpha and Hbeta lines respectively.
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