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Let $G$ be a group which is a finite extension of the group $H$.

As far as I know, by definition, this means that there is a finite normal subgroup $N$ of $G$ such that $G/N=H$.

But, is it equivalent to say that there exists a finite index subgroup of $G$ which is isomorphic to $H$?

Moreover, I'm interested in the following: Assume that $N$ is a finite normal subgroup of $G$ and that $Z^d$ has finite index in $G/N$. Is $G$ a finite extension of $Z^d$?

So far, both questions have been answered negatively. What about the following:

If $G$ is a finite extension of $H$, does there exist a finite index subgroup of $G$ which is isomorphic to $H$? Secondly, if $N$ is a finite normal subgroup of $G$ and $Z^d$ has finite index in $G/N$, does there exist a finite index subgroup of $G$ which is isomorphic to $Z^d$?

2 Answers 2