References
[1] Montalban
Antonio Montalbán. Computable Structure Theory: Within the Arithmetic. Perspectives in Logic, Cambridge University Press, 2021. For the cited numbering, see the author's draft of June 25, 2019, Example III.39 and the following paragraph, p. 50. https://math.berkeley.edu/~antonio/CSTpart1.pdf.
[2] Calvert
Wesley Calvert, Valentina Harizanov, and Alexandra Shlapentokh. Turing degrees of isomorphism types of algebraic objects. Journal of the London Mathematical Society 75(2) (2007), 273–286. doi:10.1112/jlms/jdl012. Preprint: arXiv:math/0507128.
[3] Fuchs
László Fuchs. Abelian Groups. Springer Monographs in Mathematics, Springer, 2015. doi:10.1007/978-3-319-19422-6. General background for the terminology of pure subgroups and mixed abelian groups. The arguments used here are proved in the text.
[4] Lupini
Martino Lupini, Alexander Melnikov, and André Nies. Computable topological abelian groups. arXiv:2105.12897v2, 2021. Definition 3.3 supplies the computable inverse-system convention. Section 3.2 describes symmetric cocycles and abelian extensions.
[5] Wilkes
Gareth Wilkes. Part III Profinite Groups. Lecture notes, Lent Term 2020. Sections 1.2 and 3.1 discuss profinite completion and its functoriality. https://www.dpmms.cam.ac.uk/~grw46/LectureNotes.pdf.