Is the proper class of all sets with binary operation being symmetric difference and identity being $\varnothing$ an element of $\textbf{Grp}$?

Question

My initial impression is yes because the construction fulfills the first-order axiomatization of a group; however, if this was the case then the forgetful functor from $\textbf{Grp} \to \textbf{Set}$ would not be well-defined on this object. This leads me to wonder if $\textbf{Grp}$ is a subcategory of a larger category of groups, where $\textbf{Grp}$ are groups which are simultaneously sets.