This is a general question:
Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original Leray Hirsch theorem for obvious reasons like fibre are not connected and relative dimension $0$ e.t.c.
Also as we know, Grothendieck's way to define Chern classes of complex vector bundle $E$ of rank $r$ by the equation $u^r=c_1(E)u^{r-1}+\ldots+(-1)^rc_r(E)$, where $u$ is first chern class of tautological line bundle of the projective bundle $\mathbb{P}(E)$, this is purely from Leray Hirsch theorem.
As for principal $G$-bundle $f: X\rightarrow Y$, where $X,Y$ are smooth projective varieties, $G$ finite. for example $G=\mathbb{Z}/2$. we can compute the cohomology ring of Eilenberg-Maclane space $K(\mathbb{Z}/2,1)=\mathbb{RP}^\infty$. Take the unique generator for $H^{2i}(K(\mathbb{Z}/2,1),\mathbb{C}))$, then we get characteristic classes $\eta_i(X)$ of $G$-bundle $X$.
My question: Are there results that tell us the relation between the first Chern class of $\mathcal{O}_X(-1)$ and the characteristic classes $\eta_i(X)$?
($\mathcal{O}_X(-1)$ replaces the tautological bundle of the projective bundle $\mathbb{P}(E)$)