Let $M$ and $E$ be (topological) manifolds with boundaries $\partial M$ and $\partial E$ respectively and assume we have a finite-sheeted covering $\rho: \partial E\to \partial M$. Is it possible to find an extension $\tilde{\rho}:E\to M$ of $\rho$ which is a covering projection?
I am particularly interested in the case that $M$ and $E$ are $3$-dimensional manifolds and $\rho$ is $2$-sheeted.
Certainly not in general. For instance, if $\partial M=\partial E$ and $\rho$ is the identity map, this would be saying that any two connected manifolds with the same boundary are homeomorphic. This would imply that any two connected manifolds without boundary of the same dimension are homeomorphic, since you can remove a ball from each of them and get a homeomorphism, and then glue the ball back in.
[I am implicitly assuming here that you want the boundary to be nonempty and for $M$ and $E$ to be connected. If you allow the boundary to be empty, you get that any manifold without boundary can cover any other, which is also certainly false.]