So the fact that you had a hard time thinking of cosplit sequences of groups and the last question got me thinking (along the lines of Joel's comment actually)... what I came up with is probably standard (to people who well know it).

Suppose the sequence A-> B-> C is an exact sequence of groups. Then if it splits B is a quotient of the free product A*C (the coproduct in Grp) and if it cosplits B is a subgroup of the product AxC.
The idea (in the splitting case - this is enough since they are dual) is to use the universal property + the splitting to get a map from the coproduct. Then use the factorizations + exactness + element chase to check it is an epi in terms of right cancellation. There might be a slicker way to do this, but a way to get it for free from universal properties didn't occur to me.

This gives a philosophical explanation of why split sequences are easy to find, they are given by a presentation for B in terms of A and C (e.g. your dihedral group example). I think cosplitting seems a bit weirder since one is defining a group as a subgroup of a product - is this less natural to people who actually do group theory?

I haven't checked but I suspect that one cannot have a split and cosplit sequence in Grp - it seems like the fact that there is no biproduct should be an obstruction to this based on the above but I am not sure. The largest class of categories which springs to mind where Andrew's trick works would be quasi-abelian categories for strict exact sequences (so pretty much exact categories).

Finally I thought I'd point out some whackier examples where one has splitting iff cosplitting behaviour. In a triangulated category one has that every monomorphism is split and every epimorphism is split - in particular a triangle "splits" iff it "cosplits". The same is actually true for "exact sequences of triangulated categories". A fully faithful exact functor S -> T admits a right adjoint (cosplitting) iff T -> T/S admits a right adjoint (splitting).

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So the fact that you had a hard time thinking of cosplit sequences of groups and the last question got me thinking (along the lines of Joel's comment actually)... what I came up with is probably standard (to people who well know it).

Suppose the sequence A-> B-> C is an exact sequence of groups. Then if it splits B is a quotient of the free product A*C (the coproduct in Grp) and if it cosplits B is a subgroup of the product AxC. This gives a philosophical explanation of why split sequences are easy to find, they are given by a presentation for B in terms of A and C (e.g. your dihedral group example). I think cosplitting seems a bit weirder since one is defining a group as a subgroup of a product - is this less natural to people who actually do group theory?

I haven't checked but I suspect that one cannot have a split and cosplit sequence in Grp - it seems like the fact that there is no biproduct should be an obstruction to this based on the above but I am not sure. The largest class of categories which springs to mind where Andrew's trick works would be quasi-abelian categories for strict exact sequences (so pretty much exact categories).

Finally I thought I'd point out some whackier examples where one has splitting iff cosplitting behaviour. In a triangulated category one has that every monomorphism is split and every epimorphism is split - in particular a triangle "splits" iff it "cosplits". The same is actually true for "exact sequences of triangulated categories". A fully faithful exact functor S -> T admits a right adjoint (cosplitting) iff T -> T/S admits a right adjoint (splitting).