Can I define a group $F$ that is free on the proper class $S$ of all sets? If this construction were possible, F would presumably be the largest free group in the sense that every other free group would be embedded in it.
If it isn't possible to construct the largest free group, can I construct a free group that is maximal, in the sense that it can not be embedded in any other free group? Notice that if we order the class $\mathcal{F}$ of all free groups with the order $F \leq F^\prime$ iff there exists an embedding $F \rightarrow F^\prime$, then every chain in $\mathcal{F}$ will have an upper bound, whereby we can abuse Zorn's Lemma to get a maximal element.
I know such a group will not exist in Zermelo-Fraenkel Set Theory, hence, for lack of a better word I called this group a "proper group". Nevertheless, is there a alternative to Zermelo-Fraenkel in which this large group I am trying to construct exists? If so, how can I construct it?