Given two smooth three-manifolds $(M,\partial M)$ and $(N,\partial N)$ with smooth boundary, if we know that there is a covering map $\Gamma:\partial M\to\partial N$, is there a simple criterion to check whether $\Gamma$ extends to a (possibly branched) covering map $\gamma:(M,\partial M)\to (N,\partial N)$? Perhaps a criterion on the induced maps between various homology groups?
Furthermore, if such an extension does exist, is it true that the natural homomorphism $\text{Deck}(M\to N)\to\text{Deck}(\partial M\to\partial N)$ between the respective groups of Deck transformations is surjective?