For (b) recall that a prime ideal is the kernel of a map from $G$ into a field. Such a field must be an extension field of $H$ and the image of $G$ consists of $H$ and the $p^{r_i}$-th roots of certain elements $a_i$ of $k$ where $p$ is the characteristic. Thus the image of $G$ is a purely extension of $H$ and so embeds in the perfect closure $H^i$ of $H$. Now see that there is only one such map from $G$ extending the identity on $H$.

For (b) recall that a prime ideal is the kernel of a map from $G$ into a field. Such a field must be an extension field of $H$ and the image of $G$ is generated by $H$ and the $p^{r_i}$-th roots of certain elements $a_i$ of $k$ where $p$ is the characteristic. Thus the image of $G$ is a purely inseparable extension of $H$ and so embeds in the perfect closure $H^i$ of $H$. Now see that there can only one such map from $G$ extending the identity on $H$.