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Let us say that a group $H$ is almost projective if, given any group epimorphism $f\colon G\to H$, there is an embedding $i\colon H\to G$.

Does it follow that $H$ is free? If not, is there a characterisation of such groups? Does the answer change if we restrict ourselves to the category of finitely generated groups?

Note that I do not require that $f\circ i$ be identity, so this may be strictly weaker than being truly projective. For example, all finite cyclic groups are almost projective in the category of torsion groups, even though they are not projective (unless they are trivial), and I think all finite abelian groups are almost projective in the category of torsion abelian groups.

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    $\begingroup$ Yes (in the category of all groups, or in the category of all fg groups), take $G$ free: then $H$ embeds into $G$, so $H$ is free (Nielsen-Schreier theorem). $\endgroup$
    – YCor
    Commented Feb 14 at 16:36
  • $\begingroup$ @YCor: Wow, that's embarrasing. Can you post this as an answer so that I can accept it? Or I can just post it as CW if you prefer. $\endgroup$
    – tomasz
    Commented Feb 14 at 17:52

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Yes.

As noted by YCor in the comments, both for the category of groups and the category of finitely generated grups if we take $G$ to be a free group (with the rank equal to the minimal size of a generating set for $H$), then we clearly have an epimorphism $G\to H$, so $H$ embeds into $G$, and thus it is free, as a subgroup of a free group.

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