Recovering a group from its quotient group Suppose I've a group $G$ and a normal subgroup $H$. I know the structure of $G/H$ as well as the structure of $H$. Is it possible to recover the original structure of G from this?

Unfortunately, no; the simplest example is $G_1=\mathbb{Z}/4\mathbb{Z}$ and $H_1=2\mathbb{Z}/4\mathbb{Z}$, as compared with $G_2=(\mathbb{Z}/2\mathbb{Z})\times(\mathbb{Z}/2\mathbb{Z})$ and $H_2=(\mathbb{Z}/2\mathbb{Z})\times\{\overline{0}\}$. We have
$$H_1\cong H_2\cong\mathbb{Z}/2\mathbb{Z},\qquad G_1/H_1\cong G_2/H_2\cong\mathbb{Z}/2\mathbb{Z}$$
but $G_1\not\cong G_2$. The general problem of determining what groups $G$ could arise, given a specified normal subgroup and the corresponding quotient, is called the group extension problem.
