Geodesic
Simultaneous Extension of Partial Endomorphisms of Groups
Glasgow mathematical journal, Tome 2 (1954) no. 1, pp. 37-46

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Let μ be a homomorphic mapping of some subgroup A of the group G onto a subgroup Ḃ (not necessarily distinct from A) of G; then we call μ a partial endomorphism of G. If A coincides with G, that is, if the homomorphism is defined on the whole of G, we speak of a total endomorphism; this is what is usually called an endomorphism of G. A partial (or total) endomorphism μ*extends or continues a partial endomorphism μ if the domain of μ* contains the domain of μ, that is, μ* is defined for (at least) all those elements for which μ. is defined, and moreover μ* coincides with μ where μ is defined. 
Chehata, C. G. Simultaneous Extension of Partial Endomorphisms of Groups. Glasgow mathematical journal, Tome 2 (1954) no. 1, pp. 37-46. doi: 10.1017/S2040618500032986
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[(1)] (1)Higman, G., Neumann, B. H., and Neumann, H., “Embedding theorems for groups,” J. London Math. Soc., 24 (1949), 247–254. Google Scholar | DOI

[(2)] (2)Neumann, B. H., and Neumann, Ḣanna, “Extending partial endomorphisms of groups,” Proc. London Math. Soc., (3) 2, (1952), 337–348. Google Scholar | DOI

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