Let $C_g$ be a general curve of genus $g$. If $g\ge4$ then the monodromy group of a primitive cover $C_g\to\P$ of degree $n$ is either $S_n$ or $A_n$, and both cases actually occur (under suitable conditions on $n$ for fixed $g$). For $g=3$ also the groups $GL_3(2)$ and $AGL_3(2)$ occur. In the present paper we settle the last possible case of $AGL_4(2)$. This requires new methods (which may be of independent interest) studying the combinatorial structure of degenerate covers.