@AaronMazel-Gee This would be so much easier if we were in more compatible time-zones :). Let me try to explain from the beginning

A G-category is a presheaf of categories over O_G, i.e. the category of transitive G-sets. Often I write C:O_G^{op}→Cat_∞. Often we represent them with cocartesian fibrations over O_G^{op} because that turns out to be the most technically convenient representation

For any category $\mathcal{C}$ you can form the G-category of G-objects $\underline{\mathcal{C}}_G$. This is the presheaf sending $G/H$ to $\mathrm{Fun}(O_H^{op},\mathcal{C})$

So, for example, when you feed $\mathcal{C}=\mathrm{Space}$ you get the presheaf sending $G/H$ to the category of genuine $H$-spaces

But be warned: when you feed it $\mathcal{C}=\mathrm{Spectra}$ you do not get the presheaf sending $G/H$ the category of genuine $H$-spectra. Rather you get the "naive" or "Bredon" G-spectra

Regardless, just by functoriality if $\mathcal{C}$ is a $G$-category the value $\mathcal{C}(G/H)$ has an action of the Weyl group of $H$, $W_GH=N_GH/H$

This action is often non-trivial, even for $G$-spaces. In fact morally this "twists the action" of $H$ on the objects of $\mathcal{C}(G/H)$ via the conjugation by the elements of $W_GH$. So when $G$ is abelian this action is trivial on spaces (and, I think, on spectra too), but this is by no means a universal rule

(little pause and then I elaborate on G-(co)limits)

Ok, I'm back. The story about G-(co)limits is told in Exposé II (Jay's thesis). There he proves necessary and sufficient conditions for a functor to have a (co)limit, all the details about G-Kan extensions and the whole cofinality story

In particular there is a canonical formula to compute G-(co)limits, the Bousfield-Kan formula, representing them as the geometric realization of a bunch of G-coproducts. So let's understand G-coproducts first and see what we can do

If I is a finite G-set (in fact any G-space), we can consider its G-category of points, $\underline{I}$ sending $G/H$ to $\mathrm{Hom}_G(G/H,I)=I^H$. This is of course just the restriction of the presheaf represented by $I$

If $G/H$ is a $G$-orbit (i.e. a transitive $G$-set) it turns out by some version of Yoneda that $\mathrm{Fun}_G(\underline{G/H},\mathcal{C})=\mathcal{C}(G/H)$ for every $G$-category $\mathcal{C}$

So the $G$-coproduct is (among other things) the left adjoint of the functor $\mathrm{Fun}_G(\underline{G/G},\mathcal{C})→\mathrm{Fun}_G(\underline{I},\mathcal{C})$ which under our identification is just the functor $\mathcal{C}(G/G)→\prod_i \mathcal{C}(G/H_i)$ where $I=\coprod_i G/H_i$

If we write $\mathrm{res}^G_H:\mathcal{C}(G/G)→\mathcal{C}(G/H)$ for the functor obtained by applying $\mathcal{C}$ to the arrow $G/H→G/G$ in $O_G$ and $\mathrm{ind}^G_H$ for its left adjoint, we see that the indexed coproduct $\coprod_I$ sends $\{X_i\}\in\prod_i \mathcal{C}(G/H_i)$ to $\coprod_i \mathrm{ind}_{H_i}^GX_i$ as well it should

Regarding the formula for the pushout I gave you: indeed you need an argument saying that $G$-colimits in functor $G$-categories are computed levelwise (which is true), and that formula is not a special case of a general formula. It's just that pushouts are sufficiently easy that we can manipulate them fairly concretely