  We say that a group $G$ is of \textit{profinite type} if it can be realized
as a Galois group of some field extension. Using Krull's theory, this is
equivalent to the ability of $G$ to be equipped with a profinite topology. We
also say that a group of profinite type is \textit{profinitely rigid} if it
admits a unique profinite topology. In this paper we study when abelian groups
and some group extensions are of profinite type or profinitely rigid. We also
discuss the connection between the properties of profinite type and profinite
rigidity to the injectivity and surjectivity of the cohomology comparison maps,
which were studied by Sury and other authors.
