$Qqpi$ groups and quasi-equivalence

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