Are C-rich ultra iron-poor stars also He-rich?

Context. The three most iron-poor stars presently known (Fe/H equal to -5.96, -5.4 and -4.75) are carbon-rich, so they are called C-Rich Ultra-Metal Poor Stars (CRUMPS). The origin of their peculiar surface abundances is not understood. Aims. We propose a synthetic view of the different models proposed so far to explain the peculiar abundances observed on the surface of the CRUMP stars. We deduce some expected trends based on nucleosynthetic arguments and look for signatures that allow models to be distinguished. We discuss the conditions for having CRUMP stars which are He-rich, i.e. with a mass fraction of helium greater than 0.30 and up to 0.60. Methods. We discuss the chemical composition of stars made of interstellar medium mixed with wind material of very metal-poor massive stars, with wind plus supernova ejecta and with material extracted from the envelope of early asymptotic giant branch (AGB) stars. Rotating and non-rotating models are considered.