  In this paper we continue the study of positive scalar curvature (psc)
metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular
stratum $\beta M$ (a closed manifold of positive codimension) and associated
link equal to $L$, a smooth compact manifold. We briefly call such spaces
manifolds with $L$-fibered singularities. Under suitable spin assumptions we
give necessary index-theoretic conditions for the existence of wedge metrics of
positive scalar curvature. Assuming in addition that $L$ is a simply connected
homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple
compact Lie group $G$ acting transitively on $L$ by isometries, we investigate
when these necessary conditions are also sufficient. Our main result is that
our conditions are indeed sufficient for large classes of examples, even when
$M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the
space of such psc metrics and show that it often splits into many cobordism
classes.
