  Subsequent to Paper I, the evolution and fragmentation of a rotating
magnetized cloud are studied with use of three-dimensional MHD nested-grid
simulations. After the isothermal runaway collapse, an adiabatic gas forms a
protostellar first core at the center of the cloud. When the isothermal gas is
stable for fragmentation in a contracting disk, the adiabatic core often breaks
into several fragments. Conditions for fragmentation and binary formation are
studied. All the cores which show fragmentations are geometrically thin, as the
diameter-to-thickness ratio is larger than 3. Two patterns of fragmentation are
found. (1) When a thin disk is supported by centrifugal force, the disk
fragments through a ring configuration (ring fragmentation). This is realized
in a fast rotating adiabatic core as Omega >0.2 tau_ff^-1, where Omega and
tau_ff represent the angular rotation speed and the free-fall time of the core,
respectively. (2) On the other hand, the disk is deformed to an elongated bar
in the isothermal stage for a strongly magnetized or rapidly rotating cloud.
The bar breaks into 2 - 4 fragments (bar fragmentation). Even if a disk is
thin, the disk dominated by the magnetic force or thermal pressure is stable
and forms a single compact body. In either ring or bar fragmentation mode, the
fragments contract and a pair of outflows are ejected from the vicinities of
the compact cores. The orbital angular momentum is larger than the spin angular
momentum in the ring fragmentation. On the other hand, fragments often quickly
merge in the bar fragmentation, since the orbital angular momentum is smaller
than the spin angular momentum in this case. Comparison with observations is
also shown.
