  This paper targets to generalize the notion of Hopfian groups in the
commutative case by defining the so-called {\bf relatively Hopfian groups} and
{\bf weakly Hopfian groups}, and establishing some their crucial properties and
characterizations. Specifically, we prove that for a reduced Abelian $p$-group
$G$ such that $p^{\omega}G$ is Hopfian (in particular, is finite), the notions
of relative Hopficity and ordinary Hopficity do coincide. We also show that if
$G$ is a reduced Abelian $p$-group such that $p^{\omega}G$ is bounded and
$G/p^{\omega}G$ is Hopfian, then $G$ is relatively Hopfian. This allows us to
construct a reduced relatively Hopfian Abelian $p$-group $G$ with $p^{\omega}G$
an infinite elementary group such that $G$ is {\bf not} Hopfian. In contrast,
for reduced torsion-free groups, we establish that the relative and ordinary
Hopficity are equivalent. Moreover, the mixed case is explored as well, showing
that the structure of both relatively and weakly Hopfian groups can be quite
complicated.
