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.



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



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> If a $dg$-algebra $\mathcal{A}$ with a $G$-action is formal, then it is also equivariantly formal? 
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Are any results in this direction known?



Equivariant Formality
I believe your second statement is true when $\mathcal{A}$ is the minimal Sullivan model of a $G$-space $X$. Namely, if the $G$-space $X$ is rationally formal, then it is rationally $G$-formal, in the sense that it has a rational $G$-minimal model which can be constructed from $H^\ast(X;\mathbb{Q})$ with it's $G$-action.


This is discussed in Section 3.3 of the book *Algebraic Models in Geometry* by Félix, Oprea and Tanré, see in particular Remark 3.30(2). They refer to a paper of John Oprea: *Lifting homotopy actions in rational homotopy theory*, 
J. Pure Appl. Algebra 32 (1984), no. 2, 177–190. 


