How to find the group structure of $G$ when you know $H, K$ and $G/H \simeq K$

Question

I have the following set up: $G, H, K$ are some groups, and I know $$ H = \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$$ and $$ K = \mathbb{Z}/2\mathbb{Z}. $$ I also have $$ \frac{G}{H} \simeq K $$ How do I find the group structure of $G$?

Answer 1

There's more than one such group.

Both $(\mathbb{Z}/2\mathbb{Z})^3$ and $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$ are abelian groups of order $8$ with subgroups isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. (For $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$, consider the subgroup $\{(0,0), (1,0), (0,2), (1,2)\}$)

In both cases, the quotient has size $2$ and therefore must be $\mathbb{Z}/2\mathbb{Z}$.

These two groups are not isomorphic, so you can't determine the group structure from a subgroup and quotient.