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As for a generic parameter dependent hamiltonian with the time reversal (TR)invariance, a non Abelian Berry connection with the Kramers (KR) degeneracy areintroduced by using a quaternionic Berry connection. This quaternionicstructure naturally extends to the many body system with the KR degeneracy. Itstopological structure is explicitly discussed in comparison with the onewithout the KR degeneracy. Natural dimensions to have non trivial topologicalstructures are discussed by presenting explicit gauge fixing. Minimum models tohave accidental degeneracies are given with/without the KR degeneracy, whichdescribe the monopoles of Dirac and Yang. We have shown that the Yang monopoleis literally a quaternionic Dirac monopole. The generic Berry phaseswith/without the KR degeneracy are introduced by the complex/quaternionic Berryconnections. As for the symmetry protected $\mathbb{Z}_2$ quantization of thesegeneral Berry phases, a sufficient condition of the $\mathbb{Z}_2$-quantizationis given as the inversion/reflection equivalence. Topological charges of theSO(3) and SO(5) nonlinear $\sigma $-models are discussed in their relation tothe Chern numbers of the $CP^1 $ and $HP^1$ models as well.

Symmetry protected Z2-quantization and quaternionic Berry connection with Kramers degeneracy