Let C be an affine curve over an algebraically closed field k of characteristic \(p>0\) . Given an embedding problem \((\beta :\Gamma \longrightarrow G, \alpha : \pi ^{\acute{\textrm{e}}\textrm{t}}_1(C)\longrightarrow G)\) for \(\pi _1^{\acute{\textrm{e}}\textrm{t}}(C)\) where \(\beta \) is a surjective homomorphism of finite groups with prime-to-p kernel H, we discuss when an H-cover of the G-cover of C corresponding to \(\alpha \) is a solution. When H is abelian and G is a p-group, some necessary and sufficient conditions for the solvability of the embedding problems are given in terms of the action of G on a certain generalization of m-torsion of the Picard group.
