Consider a group $G$. We can get to a monoidal category $\cal G$ with objects being the group elements, tensor product given by group multiplication and just the identity morphisms as morphisms.
Further, let $\cal C$ be a monoidal category.
Assume we have an injective group homomorphism $$G\to\text{Aut}(\cal C)$$ into the group of monoidal autoequivalences of $\cal C$.
How do we get a monoidal functor ${\cal G}\to\text{Aut}(\cal C)$? Here the tensor product on $\text{Aut}(\cal C)$ is given by composition.