  In this paper we study modular extendability and equimodularity of
endomorphisms and E$_0$-semigroups on factors with respect to f.n.s. weights.
We show that modular extendability is a property that does not depend on the
choice of weights, it is a cocycle conjugacy invariant and it is preserved
under tensoring. We say that a modularly extendable E$_0$-semigroup is of type
EI, EII or EIII if its modular extension is of type I, II or III, respectively.
We prove that all types exist on properly infinite factors.
  We also compute the coupling index and the relative commutant index for the
CAR flows and $q$-CCR flows. As an application, by considering repeated tensors
of the CAR flows we show that there are infinitely many non cocycle conjugate
non-extendable $E_0$-semigroups on the hyperfinite factors of types II$_1$,
II$_{\infty}$ and III$_\lambda$, for $\lambda \in (0,1)$.
