Every injective endomorphism arises as the restriction of an inner automorphism

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Statement

Suppose G is a group and \sigma is an Injective endomorphism (?) of G. Then, there exists a group K containing G as a subgroup, as well as an Inner automorphism (?) \alpha of K, such that the restriction of \alpha to G equals \sigma.

Notice that any endomorphism that arises as the restriction of an inner automorphism must be injective, hence this result is tight.

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