Classify all $3$-manifolds such that this map is injective

Question

Is it possible to classify all the compact, connected and orientable $3$-manifolds $M$ with nonempty boundary such that the map $H_2(M, \partial M) \to H_1(\partial M)$, appearing in the long exact sequence of the pair $(M, \partial M)$, is injective? Obviously, a sufficient condition is that $H_2(M) = 0$, but can we describe the manifolds in the general case?