In Wiki, under the item "category of groups", it states that the snake lemma fails in category of groups, however the nine lemma is valid. However, in the preface of the book " Mal'cev, protomodular, homological and semi-abelian categories ", it says "And category theory could not grasp either the conceptual foundations of the homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid and strongly meaningful in the category Gp of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way."


Does the snake lemma fail or not in the category of groups? Any counter example or reference concerning that?



snake lemma in category of groups
The snake lemma doesn't make sense in the category of groups in general, because maps don't always have cokernels. If you assume that the three vertical maps in your snake lemma diagram do have cokernels, that is, that their images are normal subgroups of their targets, then the snake lemma is true with the same proof as in the abelian category case (replacing the $0$'s by $1$'s and the minus signs by inversions, as J. Polak pointed out). 


