Do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally isomorphic to the underlying set functor?

Question

I'm trying to get some intuition for the notion of natural isomorphism.

To that end, my question is: do there exist faithful functors $\mathrm{Grp} \rightarrow \mathrm{Set}$ that aren't naturally isomorphic to the underlying set functor?

Thanks.

Answer 1

Sure, the functor $F(G) = G\cup \{*\}$ (underlying set plus an extra point). For $\phi: G\to H$, take $F(\phi):G\cup \{*\}\to H\cup\{*\}$ defined by $F(\phi)(g) = \phi(g)$ for $g\in G$ and $F(\phi)(*) = *$.

More generally, take any faithful functor $\text{Set}\to \text{Set}$ which isn't isomorphic to the identity and compose it with the underlying set functor.