Supermassive stars (SMSs) forming via very rapid mass accretion (Ṁ* ≳ 0.1M⊙ yr-1) could be precursors of supermassive black holes observed beyond a redshift of about six. Extending our previous work, here we study the evolution of primordial stars growing under such rapid mass accretion until the stellar mass reaches 104-5M ⊙. Our stellar evolution calculations show that a star becomes supermassive while passing through the "supergiant protostar" stage, whereby the star has a very bloated envelope and a contracting inner core. The stellar radius increases monotonically with the stellarmass until≃100 AU form* ≳ 104M⊙, after which the star begins to slowly contract. Because of the large radius, the effective temperature is always less than 104 K during rapid accretion. The accreting material is thus almost completely transparent to the stellar radiation. Only for M* ≳ 105M⊙ can stellar UV feedback operate and disturb the mass accretion flow. We also examine the pulsation stability of accreting SMSs, showing that the pulsation-driven mass loss does not prevent stellar mass growth. Observational signatures of bloated SMSs should be detectable with future observational facilities such as the James Webb Space Telescope. Our results predict that an inner core of the accreting SMS should suffer from the general relativistic instability soon after the stellar mass exceeds 10 5M⊙. An extremely massive black hole should form after the collapse of the inner core.
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