As requested, here is an answer summarizing axioms for the category of groups that were [given][1] by Pierre Leroux, and which I learned from an MSE answer of Arnaud D. The category of groups is the unique category $C$ with the following properties: 

 1. It has all limits and colimits, and a zero object.
 2. It has as a full subcategory $C_Z$ a category closed under coproducts and containing a cogroup $Z,$ and generated by the morphisms required by those properties. 
 3. $C_Z$ is closed under subobjects in $C.$
 4. $Z$ is a regular-projective generator of $C.$
 5. Every inclusion of the equivalence class of $0$ in an equivalence relation on an object of $C$ is a normal monomorphism.
 6. Every object of $C$ is a subobject of a simple object. 


  [1]: https://www.cambridge.org/core/journals/canadian-mathematical-bulletin/article/une-caracterisation-de-la-categorie-des-groupes/BDD4B7EA2FD122F7E0F1C81236162FE4