# Embedding in a finitely generated group, up to finite index

Suppose that $G$ is a group with a finite subgroup $H$ such that $G/H$ embeds into a finitely generated group. Does $G$ itself embed into a finitely generated group?

• $G/H$ is a finite group, hence it trivially embeds in a finitely generated group (in itself). This does not give any information. – Crostul Jan 26 '17 at 20:10
• Oops, sorry, $H$ is supposed to be finite, not finite index! (Question edited.) – Mark Jan 26 '17 at 20:21
• If you want a direct proof, you could embed $G$ in the the wreath product $H \wr G/H$, which embeds in $H \wr N$ when $G/H < N$, and $H \wr N$ is finitely generated if $N$ is. – Derek Holt Jan 26 '17 at 21:18