We discuss in detail the evolutionary properties of low-mass stars (M < 1 M sub(o)) having metallicity lower than Z = 10 super(-6) from the pre-main sequence up to (almost) the end of the early asymptotic giant branch phase. We also discuss the possibility that the large C/Fe, N/Fe observed on the surface of the most iron-poor star currently known, HE 0107-5240, could be attributed to the autopollution induced by the penetration of the He convective shell into the H-rich mantle during the He core flash of a low-mass, very low metallicity star. On the basis of a quite detailed analysis, we conclude that the autopollution scenario cannot be responsible for the observed chemical composition of HE 0107-5240.