  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.
