Let $Y_1$ and $Y_1'$ be index two degree one Fano threefolds. Suppose we have an Fourier-Mukai equivalence $\Phi_P : \mathrm{D}^b(Y_1) \to \mathrm{D}^b(Y_1')$. Can anything be said about the kernel $P$, i.e. is there some kind of a classification in this Fano case? Thank you.