  The category $\mathrm{FI}_G$ was first defined and explored by Sam-Snowden.
Here, we develop more of the machinery of $\mathrm{FI}_G$-modules and find
numerous examples to apply it to, extending the work of Church-Ellenberg-Farb
and Wilson.
  In particular we develop a notion of character polynomials for
$\mathrm{FI}_G$-modules with $G$ finite, a notion of representation stability
which we call $K_0$-stability even when $G$ is infinite virtually polycyclic,
and apply the notion of finite presentation degree when $G$ is a general
infinite group. We use this to analyze numerous families of $(G^n \rtimes
S_n)$-modules, such as:
  -the cohomology and homotopy groups of orbit configuration spaces
  -the diagonal coinvariant algebra of complex reflection groups
  -the homology of affine pure braid groups of type $\widetilde{A}_n$ and
$\widetilde{C}_n$
  -the cohomology of Fouxe-Rabinowitsch groups
  and many more examples.
