# K theoretic pushforward along gerbes

I have a nontrivial gerbe $$\pi : \mathscr{G} \to X$$ banded by a cyclic group $$G = \mathbb{Z}/r$$. I'm working over $$\mathbb{C}$$. I want to describe $$\pi_\ast$$ and relate the fundamental class $$[\mathcal{O}_X]$$ with the pushforward of the fundamental class $$[\mathcal{O}_{\mathscr{G}}]$$ in $$K$$ theory. If it helps, $$X$$ is a smooth algebraic stack.

In chow with $$\mathbb{Q}$$ coefficients, one has $$\pi_\ast [\mathscr{G}] = \dfrac{1}{r} [X]$$ because it is a morphism proper of pure degree $$\dfrac{1}{r}$$. If $$\mathscr{G} = BG_X$$ is a trivial gerbe over $$X$$, the sequence $$X \overset{p}{\to} BG_X \overset{\pi}{\to} X$$ and the formula for pushforward for torsors $$p$$ gives $$p_\ast[\mathcal{O}_{X}] = [\mathcal{O}_X^{\oplus \#G}],$$ where the latter sheaf has $$G$$ action permuting the coordinates (if I have this right). I'm not sure how to compare this with $$[\mathcal{O}_{BG_X}]$$, which I believe is $$[\mathcal{O}_X]$$ with the natural $$G$$ action. I am then told that $$\pi_\ast$$ takes $$G$$-invariant sections.

I can live with inverting a few numbers, $$K() \otimes \mathbb{Z}[\dfrac{1}{r_1 r_2 \dots, r_k}]$$ if necessary to get a relationship. I don't want to just tensor with $$\mathbb{Q}$$ because then $$K$$ theory is basically just chow.

Questions:

1. What is the pushforward $$[\mathcal{O}_{\mathscr{G}}]$$?
2. How is it related to $$[\mathcal{O}_X]$$? Is there a natural variant of the fundamental class of $$\mathscr{G}$$ that pushes forward precisely to that of $$X$$? Something like $$r[\mathscr{G}]$$ in chow?
3. Do I have the above right? What does this look like in the example of a trivial gerbe?
4. References for $$K$$ theory on gerbes would be greatly appreciated.