Recovering group from factor group and factor If I know what the groups $G/N$ and $N$ are, is there anyway for me to recover the group $G$?
 A: Only with the information about $N$ and $G/N$ you can't recover the group even if you assume $G$ is abelian.
Given two abelian groups $A$ and $B$, one can define a group $\operatorname{Ext}(B,A)$ that classifies, up to isomorphisms, the (abelian) groups $G$ such that there exists an exact sequence $1\to A\to G\to B\to 1$ and, generally, this group is non zero. So, without any other information, there can be non isomorphic groups having “the same subgroup” and “the same factor group”. Consider $A=B=\mathbb{Z}/2\mathbb{Z}$; you can have $G=\mathbb{Z}/4\mathbb{Z}$ or $G=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.
If either $A$ is divisible or $B$ is free, then $\operatorname{Ext}(B,A)$ is the one element group and the structure can be recovered.
If non abelian groups are involved, the situation is even worse. The group $Q_8$ of elementary quaternions is not abelian and has a subgroup isomorphic to $\mathbb{Z}/2\mathbb{Z}$, with factor group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$.
