  We show that, for many Lie superalgebras admitting a compatible
$\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between
simple supermodules over a Lie superalgebra and simple supermodules over the
even part of this Lie superalgebra. This reduces the classification problem for
the former to the one for the latter. Our result applies to all classical Lie
superalgebra of type $I$, in particular, to the general linear Lie superalgebra
$\gl(m|n)$. In the latter case we also show that the rough structure of simple
$\gl(m|n)$-supermodules and also that of Kac supermodules depends only on the
annihilator of the $\mf{gl}(m)\oplus \mf{gl}(n)$-input and hence can be
computed using the combinatorics of BGG category $\mathcal{O}$.
