According to the "QFT Nutshell" page 102-104 about Time reversal, the author said about applying the time reversal to the spin 1/2 system.
Let $T=UK$, where $K$ complex conjugates everything to its right and $U$ is a unitary operator. Acting with $T$ on the spin-up state, we obtain the spin-down state. Thus, we need a nontricial matrix $U=\eta\sigma_2$ to flip the spin. Note that $T^2=\eta\sigma_2 K\eta \sigma_2 K = \eta\sigma_2\eta^*\sigma_2^*KK=-1$. This is the origin of Kramer's degeneracy
My question is that, it seems we can choose a different choice of the time reversal to be $U=\eta\sigma_1$ equally well. This also clearly flips the spin-up state to the spin-down state and vice versa. With this choice, we have
$$T^2=\eta\sigma_1 K\eta \sigma_1 K = \eta\sigma_1\eta^*\sigma_1^*KK=1$$
because $\sigma_1$ is real. We won't see the Kramer degeneracy anymore. But physics should not depend on the choice. Is there any other constrains we have to impose in order to get $U=\eta\sigma_2$?