  In this paper, we provide examples of Sarkisov links of type II between
complex projective Fano threefolds $X$ with $\rho(X) = 1$. To show examples of
these links, we study smooth weak Fano threefolds with extremal rays of type
$E$. We further assume that the pluri-anticanonical morphism contracts only a
finite number of curves. We numerically classify smooth weak Fano threefolds
with $\rho(X) = 2$ having an extremal ray of type $E$ both before and after a
flop as well as prove the existence of some of these numerical cases.
