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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 $\mathrm{Aut}(M^{\rm gp})$?

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    @ullo That last question isn't even about the automorphism groups. As I've said, you really need to clari$f$y your question.2012-06-27

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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.