let $M$ be a finitely generated, commutative monoid.
What are in general the relations between $\mathrm{Aut}(M)$ and $\mathrm{Aut}(M^{\rm gp})$?
When is it true that $\mathrm{Aut}(M)$ determines $\mathrm{Aut}(M^{\rm gp})$?
let $M$ be a finitely generated, commutative monoid.
What are in general the relations between $\mathrm{Aut}(M)$ and $\mathrm{Aut}(M^{\rm gp})$?
When is it true that $\mathrm{Aut}(M)$ determines $\mathrm{Aut}(M^{\rm gp})$?
I don't think there's much to say in general. For example, if every element of $M$ is idempotent ($m^2 = m$) then $M^{gp}$ is trivial. To generate examples of this you can take any finite lattice with join as the monoid operation.
Of course since $M^{gp}$ is a functor every automorphism $M \to M$ induces an automorphism $M^{gp} \to M^{gp}$. But my point is this need not be injective.