We used to think that $T^2=-1$ is for a fermionic system, spin half-integer, and there is a Kramers doublet for each energy eigenstate - Kramers theorem.
However, some context in Condensed Matter literature uses $T^2=\mp 1$ to label a bosonic particle. So I wonder, how are the following relations are related to each other?
(1) $T^2=\mp 1$,
(3) spin half-integer/integer,
(4) Kramers doublet/singlet