correction on category of groups

In an earlier post today I misdescribed a way of axiomatizing the category of groups by the triple for groups over sets. The point is that you can axiomatize the category of sets and the Eilenberg-Moore category for the triple for groups over it, and then identify the category of sets with the non-full subcategory of free groups and homomorphisms taking generators to generators; so that in a very narrow sense you would "only be talking about groups and homomorphisms". But really this amounts to defining groups as structured sets. What I want to know is, are there known axioms approaching the category of groups directly.