Extending Quotients of Knot Groups over Surfaces in $B^4$

Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $\rho\colon \pi_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$ with $\partial F = K$ over whose exterior $\rho$ extends surjectively. Equivalently, we determine whether the cover of $S^3$ branched over $K$ and induced by $\rho$ bounds a connected cover of $B^4$ branched along such a surface. When $G$ is a dihedral group, we show the obstruction can be computed by evaluating the Seifert form of $K$ on a single curve, a so-called characteristic knot associated to $\rho$. When the dihedral obstruction vanishes, we construct the surface $F$ explicitly.