This reviewer felt a pang of sadness when receiving this beautiful paper, as the first author, Fritz Grunewald, died untimely in 2010. It is well known that two non-isomorphic (residually finite) groups may have isomorphic profinite completions. Given a family \(\mathcal C\) of groups, and a group \(G\in\mathcal C\), the authors define the genus \(\mathbf g(\mathcal C,G)\) as the set of isomorphism classes of groups in \(\mathcal C\) which share the same profinite completion with \(G\). Several problems are formulated. Problem \textbf{PI} asks for interesting families \(\mathcal C\) such \(\mathbf g(\mathcal C,G)\) is always finite, and for formulas for its size. \textbf{PII} asks for families in which the genuses contain one element. \textbf{PIII} asks for families which contain groups whose genus has finitely many, but more than one element, possibly with the genus being unbounded over the family. \textbf{PIV} asks for algorithms to decide whether two finitely presented groups in a given family have isomorphic profinite completions, and \textbf{PV} for algorithms to count the number of elements in a genus. Section~2 is motivated by examples by \textit{J.-P.~Serre} [Proc. Natl. Acad. Sci. USA 47, 108-109 (1961; Zbl 0100.16701)] about topological fundamental groups of the complex points of connected smooth algebraic projective varieties. Here, \textbf{PI} is solved for Abelian-by-finite groups. Then, motivated by a letter (which is reproduced in an appendix) of the late \textit{K.~W.~Gruenberg}, the class \(\mathcal{VF}\) of finitely generated, virtually free groups is considered. The authors first show that \textbf{PII} has no general positive answer here, but then prove a one-in-a-class result under certain conditions. In Sections 3 and 4 a positive answer to \textbf{PI} is provided for \(\mathcal{VF}\). There is much more in the paper, in particular about algorithms that give positive answers to \textbf{PIV} for various subclasses of \(\mathcal{VF}\). Also, \textbf{PIV} appears to be analogous to a problem in number theory studied by \textit{J.~Ax} [see e.g.~Ann. Math. (2) 85, 161-183 (1967; Zbl 0239.10032)] and S.~Kochen. We refer to the lucidly written paper for further details.