$T^2=\mp 1$, fermion/boson, spin half-integer/integer, Kramers doublet/singlet

Question

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) $T^2=\mp 1$,

(2) fermion/boson,

(3) spin half-integer/integer,

(4) Kramers doublet/singlet