Our main goals are to get a deeper insight into the evolution and final fates of intermediate-mass, extremely metal-poor (EMP) stars. We also aim to investigate their C, N, and O yields. Using the Monash University Stellar Evolution code we computed and analysed the evolution of stars of metallicity Z = 10^-5 and masses between 4 and 9 M_sun, from their main sequence until the late thermally pulsing (super) asymptotic giant branch, TP-(S)AGB phase. Our model stars experience a strong C, N, and O envelope enrichment either due to the second dredge-up, the dredge-out phenomenon, or the third dredge-up early during the TP-(S)AGB phase. Their late evolution is therefore similar to that of higher metallicity objects. When using a standard prescription for the mass loss rates during the TP-(S)AGB phase, the computed stars lose most of their envelopes before their cores reach the Chandrasekhar mass, so our standard models do not predict the occurrence of SNI1/2 for Z = 10^-5 stars. However, we find that the reduction of only one order of magnitude in the mass-loss rates, which are particularly uncertain at this metallicity, would prevent the complete ejection of the envelope, allowing the stars to either explode as an SNI1/2 or become an electron-capture SN. Our calculations stop due to an instability near the base of the convective envelope that hampers further convergence and leaves remnant envelope masses between 0.25 M_sun for our 4 M_sun model and 1.5 M_sun for our 9 M_sun model. We present two sets of C, N, and O yields derived from our full calculations and computed under two different assumptions, namely, that the instability causes a practically instant loss of the remnant envelope or that the stars recover and proceed with further thermal pulses. Our results have implications for the early chemical evolution of the Universe.
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