Let $G$ be a group and $X$ a simply connected $G$-space. For a $K(G,1)$ space $K$ with universal cover $\widetilde{K}\rightarrow K$ we have that $G$ acts on $\widetilde{K}$ via the unique homotopy lifting property. Then on the product $\widetilde{K}\times X$ we obtain an action by $(g,(k,x))\mapsto (kg^{-1},gx)$.
I aim to express the groups $\pi_n((\widetilde{K}\times X)/G)$ in terms of the groups $G$ and $\pi_n(X)$. To me it seems that such a relation should be obtainable using a suitably chosen fibration and then consider the long exact sequence of that. One of my (failed) attempts was the following: With base space $(\widetilde{K}\times X)/G$, then the projection $$\widetilde{K}\times X\rightarrow (\widetilde{K}\times X)/G$$ has fibre $X$ and the corresponding long exact sequence turns out as $$\pi_{n+1}((\widetilde{K}\times X)/G)\rightarrow\pi_n(X)\rightarrow\pi_n(\widetilde{K}\times X)\rightarrow\pi_n((\widetilde{K}\times X)/G)\rightarrow\pi_{n-1}(X)$$ However $\pi_n(\widetilde{K}\times X)\cong\pi_n(X)$ so this particular sequence seems like a dead end.
I have previously shown that $\widetilde{K}\rightarrow K$ is a principle $G$-bundle which actually lead me to the above fibration, but other than that, I can not see what that might be used for.
My question is this: Is my above attempt in fact a dead end? If yes, what fibration could one study to come to desired conclusion, or should I try something other than looking at fibrations?
You can try the Borel fibration $$X \to EG \times_G X \to BG.$$