Section \[braided-Gcrossed\] we define the $G$-braided center of a 2-category with an action of a group $G$. In Section \[Section-center\], we show that there is a monoidal equivalence ${{\mathcal Z}}( {{\mathcal B}}^G)\simeq {{\mathcal Z}}(\Phi)^G, $ where $\Phi:{{\mathcal B}}^G\to {{\mathcal B}}$ is the forgetful pseudofunctor. When applied to the example $({}_{{{\mathcal C}}}{\mbox{\rm Mod\,}})^G$, we recover results from [@GNN].

Acknowledgments {#acknowledgments .unnumbered}
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The work of E.B and M.M. was partially supported by CONICET, Secyt (UNC), Argentina. M.M. is grateful to the department of mathematics at Universidad de los Andes, Bogotá, where part of this work was done, for the kind hospitality.

2-categories {#Section:2-cat}
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Let us briefly recall the notion of a 2-category. For more details, the reader is referred to [@KSt; @St]. For any 2-category ${{\mathcal B}}$, the set of objects, also called *0-cells*, will be denoted by $\operatorname{Obj}({{\mathcal B}})$. The composition in each hom-category ${{\mathcal B}}(A, B)$, that is, the vertical composition of 2-cells, is denoted by juxtaposition $fg$, while the symbol $\circ$ is used to denote the horizontal composition functors $$\circ : {{\mathcal B}}(B, C) \times {{\mathcal B}}(A, B) \to {{\mathcal B}}(A, C).$$ The identity of a 0-cell $A$ is written as $I_A : A \to A$. For 