$\pi_{n}(X_{G})$ when $G$ is finite and acts freely on $X$

Question

When $X\in\mathbf{sSet}_{\ast}$ and $G$ a finite group (it may be considered a constant simplicial group) that acts freely on $X$. My question is

Can I express $\pi_{n}(X_{G})$ in terms of $\pi_{n}(X)$ and $G$?

If it does not happen, what if $X$ is $n$-connected?