As requested, here is an answer summarizing axioms for the category of groups that were given 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. EDIT: I initially gravely misunderstood this to assume that $C_Z$ is freely generated by these 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.

As requested, here is an answer summarizing axioms for the category of groups that were given 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.

As requested, here is an answer summarizing axioms for the category of groups that were given 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 $C_Z$ the free category-with-coproducts-and-a-cogroup-object. Call the cogroup $Z.$
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.

Condition 2 might come across as a bit unsatisfying, but it's perhaps not so different than the characterization of the category of sets as the cocomplete well-pointed topos, in that you can easily rephrase that characterization to involve the (improper) subcategory of coproducts of $1$ being the free category-with-coproducts. Anyway, the reader can judge.

As requested, here is an answer summarizing axioms for the category of groups that were given 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. EDIT: I initially gravely misunderstood this to assume that $C_Z$ is freely generated by these 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.