  We consider finitely presented,residually finite groups $G$ and finitely
generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$
induces an isomorphism from the profinite completion of $A$ to a direct factor
of the profinite completion of $G$. We explain why $A$ need not be a direct
factor of a subgroup of finite index in $G$; indeed $G$ need not have a
subgroup of finite index that splits as a non-trivial direct product. We prove
that there is no algorithm that can determine whether $A$ is a direct factor of
a subgroup of finite index in $G$.
