  We study the gravitational collapse and formation of the Kerr black hole from
the rotating progenitor star. We follow the evolution of black hole spin,
coupled with its increasing mass. We study the effect of different level of
rotation endowed in the progenitor's envelope, and we out some constraints on
the final black hole parameters. Our method is based on semi-analytical
computations that involve stellar-evolution models of different progenitors. We
also follow numerically the black hole evolution and spacetime metric changes
during the collapse, via General Relativistic MHD modeling. (abridged)
