  In this paper we give the complete characterization of the boundedness of the
generalized fractional maximal operator $$
  M_{\phi,\Lambda^{\alpha}(b)}f(x) : = \sup_{Q \ni x} \frac{\|f
\chi_Q\|_{\Lambda^{\alpha}(b)}}{\phi (|Q|)} \qquad (x \in {\mathbb R}^n), $$
between the classical Lorentz spaces $\Lambda^p (v)$ and $\Lambda^q(w)$ for
appropriate functions $\phi$, where $0 < p,\,q < \infty$, $0 < \alpha \le r <
\infty$, $v,w,\,b$ are weight functions on $(0,\infty)$ such that $0 < B(x): =
\int_0^x b < \infty$, $x > 0$, $B \in \Delta_2$ and $B(t) / t^{\alpha / r}$ is
quasi-increasing.
