# snake lemma in category of groups

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?

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).
First, I remind what is an exact sequence of pointed sets (see, for example, "Local fields" of Serre in the chapter about non abelian cohomology) : a pointed set is just a set $A$ with a based point $x_A$. A morphism of pointed sets is defined to be a map sending the based point to the based point. A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is said to be exact if $f(A) = g^{-1}(x_C)$.
Now let's go back to the snake Lemma. A cokernel is a pointed set with the class of $0$ as based point (I denote $Coker(f:A \rightarrow B)$ to be the left cosets $B/Im(f)$ but it also works for right cosets, and I call it cokernel even if it is NOT the cokernel in the theory of category). One can check that the Snake Lemma holds in the sense I have given above (with the same proof as Alex said).