Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.

Assume further there exists a $dg$-algebra $\mathcal{B}$ and an isomorphism $f:\mathcal{A}\to \mathcal{B}$, then you can find a $G$-action on $\mathcal{B}$ such that $f$ is equivariant.

What is if $f$ not an isomorphism but only a quasi-isomorphism. Is it possible to find a $G$-structure on $\mathcal{B}$ such that $\mathcal{A}$ and $\mathcal{B}$ are equivariantly quasi-isomorphic.

I guess no. But I have hope that following statement is true:

If a $dg$-algebra $\mathcal{A}$ with a $G$-action is formal, then it is also equivariantly formal?

Are any results in this direction known?