A criterion for solving embedding problems for the etale fundamental group of curves

Let $C$ be an affine curve over an algebraically closed field $k$ of characteristic $p>0$. Given an embedding problem $(\beta :\mathrm{\Gamma }⟶G,\alpha :{\pi }_1^{et}(C)⟶G)$ 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.