Groups of profinite type and profinite rigidity

We say that a group $G$ is of $\mathit{\text{profinite}}\mathit{\text{type}}$ 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 $\mathit{\text{profinitely}}\mathit{\text{rigid}}$ 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.