Let \(k\) be a field, \(B\) a \(k\)-curve (i.e. a smooth projective and geometrically connected \(k\)-scheme of dimension 1), \(G\) a finite group, \(f:Y\to B\) a \(k\)-\(G\)-cover of curves with group \(G\) and ramification indices \(e_1,\dots,e_r\) and let \(P\) be ant subgroup of \(G\) of index \(m.\) Assume that the branch divisor of \(f:Y\to B\) has good reduction and that \(l=ch(K)\) does not divide \((|G|,m!).\) It is shown that the order of \(\mathbb{P}^{ab}\) is bounded by a constant depending on \(m, r\) and the smallest prime \(l\) not dividing \(|G|\) of good reduction of the branch divisor. It is conjectured that the orders of \(\mathbb{P}^{ab}\) are bounded by a constant depending only on \(r\) and \(m\). The connection with other conjectures is discussed.