The answer to the following question might be obvious but I haven’t found a full proof yet (neither by myself nor in the literature). So my apologies if it is trivial. 

Let $X$ be a (for simplicity quasi-projective and non-singular) complex variety $X$ on which a finite group $G$ acts. 
Deligne has shown in Hodge III that the equivariant cohomology group $H_G^k(X,\mathbb{Q})$ for any $k$ carries a natural mixed Hodge structure (MHS). On the other hand, the Leray-Serre spectral sequence for the Serre fibration 
\begin{equation}
X \to X\times_G EG \to BG
\end{equation}
degenerates (complex topology) over $\mathbb{Q}$ and hence yields an isomorphism 

\begin{equation}
H_G^k(X,\mathbb{Q})\cong H^k(X,\mathbb{Q})^G
\end{equation}

(cf. this MO question: https://mathoverflow.net/questions/120447/equivariant-cohomology-of-finite-group-actions-and-invariant-cohomology-classes?rq=1). 
Clearly, $H^k(X,\mathbb{Q})^G$ inherits a MHS from the MHS on $H^k(X,\mathbb{Q})$. 
Now my question is: 

Are these two MHS on $H_G^k(X,\mathbb{Q})$ naturally isomorphic?
More precisely, is there a simplicial version of the above Serre fibration yielding an isomorphism $H_G^k(X,\mathbb{Q})\cong H^k(X,\mathbb{Q})^G$ of MHS?

A natural candidate for such a simplicial version is 

\begin{equation}
[X/G]_\bullet \to B_\bullet G,
\end{equation}

where I (essentially) use Deligne's notation from Hodge III. Since I'm still learning simplicial methods, I was not sure if this is maybe too naive. 

Any thoughts/references/comments are very welcome!