  Motivated by the experimental observations of resonant tunnelings in the
systems with half-integer spin, such as V$_{15}$ and Mn$_4$, we study the
mechanism of adiabatic change of the magnetization in systems with the
time-reversal symmetry. Within the time-reversal symmetric models, effects of
several types of perturbations are investigated. Although tunneling between the
ground states is suppressed in a simple Kramers doublet, we show that the
nonadiabatic transition governed by the Landau-Zener-St\"uckelberg mechanism
occurs in many cases due to the additional degeneracy of the ground state. We
also found more general cases where LZS mechanism can not be applied directly
even the system shows a kind of adiabatic change of the magnetization.
