We analyze X-ray spectra and images of a sample of Seyfert 2 galaxies that unambiguously contain starbursts, based on their optical and UV characteristics. Although all sample members contain active galactic nuclei (AGNs), supermassive black holes or other related processes at the galactic centers alone cannot account for the total X-ray emission in all instances. Eleven of the twelve observed galaxies are significantly resolved with the ROSAT HRI, while six of the eight sources observed with the lower-resolution PSPC also appear extended on larger scales. The X-ray emission is extended on physical scales of 10 kpc and greater, which we attribute to starburst-driven outflows and supernova-heating of the interstellar medium. Spectrally, a physically-motivated composite model of the X-ray emission that includes a heavily absorbed (N_H > 10^{23} cm^{-2}) nuclear component (the AGN), power-law like scattered AGN flux, and a thermal starburst describes this sample well. Half the sample exhibit iron K alpha lines, which are typical of AGNs.
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