If $N$ and $H$ are finite groups, then there exist group $G$ s.t $N$ is normal in $G$ and $G/N\cong H$ (ex. $N\times H, N\rtimes H$ etc.)
Does there exist a group $G$ s.t. $N$ is characteristic subgroup of $G$ and $G/N\cong H$?
If $N$ and $H$ are finite groups, then there exist group $G$ s.t $N$ is normal in $G$ and $G/N\cong H$ (ex. $N\times H, N\rtimes H$ etc.)
Does there exist a group $G$ s.t. $N$ is characteristic subgroup of $G$ and $G/N\cong H$?