Summary: Let $X, Y$ be Fano threefolds of Picard number one and such that the ample generators of Picard groups are very ample. Let $X$ be of index one and $Y$ be of index two. It is shown that the only morphisms from $X$ to $Y$ are double coverings. In fact nearly the whole paper is the analysis of the case where $Y$ is the linear section of the Grassmannian $G(1,4)$, since the other cases were more or less solved in another article. This remaining case is treated with the help of Debarre's connectedness theorem for inverse images of Schubert cycles.