Transfer of abelian model structures to equivariant categories and homotopy squares

Let $G$ be a finite group acting on a Grothendieck category $\mathcal{A}$ with enough projectives, such that $|G|$ is invertible in $\mathcal{A}$. We prove a general lifting theorem for abelian model structures from $\mathcal{A}$ to its equivariant category $\mathcal{A}^G$, and establish a triangle