  Let $C$ be an affine curve over an algebraically closed field $k$ of
characteristic $p>0$. Given an embedding problem $(\beta:\Gamma\longrightarrow
G, \alpha: \pi^{et}_1(C)\longrightarrow G)$ for $\pi_1^{et}(C)$ 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 problem are given in
terms of the action of $G$ on certain generalized Picard group.
