A torsion-free abelian group
G
G
is
q
p
i
qpi
if every map from a pure subgroup
K
K
of
G
G
into
G
G
lifts to an endomorphism of
G
.
G.
The class of
q
p
i
qpi
groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous
Q
q
p
i
Qqpi
groups, those homogeneous groups
G
G
such that, for
K
K
pure in
G
,
G,
every
θ
:
K
→
G
\theta :K\rightarrow G
has a lifting to a quasi-endomorphism of
G
.
G.
An irreducible group is
Q
q
p
i
Qqpi
if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A
Q
q
p
i
Qqpi
group
G
G
is
q
p
i
qpi
if and only if every endomorphism of
G
G
is an integral multiple of an automorphism. A group
G
G
has minimal test for quasi-equivalence (
m
t
q
e
)
mtqe)
if whenever
K
K
and
L
L
are quasi-isomorphic pure subgroups of
G
G
then
K
K
and
L
L
are equivalent via a quasi-automorphism of
G
.
G.
For homogeneous groups, we show that in almost all cases the
Q
q
p
i
Qqpi
and
m
t
q
e
mtqe
properties coincide.