Group actions on monoidal triangulated categories and Balmer spectra

Let $G$ be a group acting on a left or right rigid monoidal triangulated category $\mathbf{K}$ which has a Noetherian Balmer spectrum. We prove that the Balmer spectrum of the crossed product category of $\mathbf{K}$ by $G$ is homeomorphic to the space of $G$-prime ideals of $\mathbf{K}$, give a concrete description of this space, and classify the $G$-invariant thick ideals of $\mathbf{K}$. Under some additional technical conditions, we prove that the Balmer spectrum of the equivariantization of $\mathbf{K}$ by $G$ is also homeomorphic to the space of $G$-prime ideals. Examples of stable categories of finite tensor categories and perfect derived categories of coherent sheaves on Noetherian schemes are used to illustrate the theory.