CatGloss

For any group $G$, we may define other groups:

1. the center $Z(G) = { h \in G \mid hg = gh\, \forall g \in G}$, a subgroup of $G$,
2. the commutator subgroup $C(G)$, the subgroup of $G$ generated by elements $ghg^{-1}h^{-1}$ for any $g,h \in G$, and
3. the automorphism group $\mathrm{Aut}(G)$, the group of isomorphisms $\phi \colon G \to G$ in $\textup{\textsf{cat}}$. Trivially, all three constructions define a functor from the discrete category of groups (with only identity morphisms) to $\textup{\textsf{cat}}$. Are these constructions functorial in
4. the isomorphisms of groups? That is, do they extend to functors $\textup{\textsf{cat}}_\text{iso} \to \textup{\textsf{cat}}$?
5. the epimorphisms of groups? That is, do they extend to functors $\textup{\textsf{cat}}_\text{epi} \to \textup{\textsf{cat}}$?
6. all homomorphisms of groups? That is, do they extend to functors $\textup{\textsf{cat}} \to \textup{\textsf{cat}}$?

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