Summary: Let \(C_g\) be a general curve of genus \(g\). If \(g\geq4\) then the monodromy group of a primitive cover \(C_g\to\mathbb P^1\) 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 \(\text{GL}_3(2)\) 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.