  We fix a finitely presented group $Q$ and consider short exact sequences
$1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$
induces a morphism of profinite completions $\hat N\to \hat G$. We prove that
this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect
and has no proper subgroups of finite index.
  We prove that there is no algorithm that, given a finitely presented,
residually finite group $G$ and a finitely presentable subgroup $P\subset G$,
can determine whether or not $\hat P\to\hat G$ is an isomorphism.
