Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $\tilde{\rho}:\pi_1(M)\rightarrow SL_2\mathbb{C}$ so that the restriction of $\tilde{\rho}$ to $\pi_1(F)$ is equal to $\rho$?

If not what is there an obstruction that allows you to identify the representaitons that do extend?