Let $X$ be a smooth, projective complex variety and $j \colon D \hookrightarrow X$ a smooth divisor. Then we have a Gysin morphism in singular cohomology


$$
j\_\ast \colon H^{\bullet}(D) \to H^{\bullet+2}(X)
$$


Now assume that $X$ is acted upon by a finite group $G$ and that $D$ is stable under this action. Then we get actions $g^\ast$ on the cohomology of $D$ and $X$. 


*Is is true that $j\_\ast g^\ast=g^\ast j\_\ast$ for any $g$ in $G$?* 


This looks like some projection formula, but I'm unable to prove it. 



Is the Gysin morphism equivariant?
Let me expand my comment slightly into an answer. To simplify matters, suppose that the coefficients are $\mathbb{Q}$ or $\mathbb{C}$. The Gysin map is Poincaré dual to the restriction map $j^\*$. In more detail, we have isomorphisms
$H^i(X)\cong H^{2n-i}(X)^\vee$ and $H^i(D)\cong H^{2n-2-i}(D)^\vee$, where $n=\dim X$, given by Poincaré pairings $(\alpha,\beta) =\int\_X\alpha\cup\beta$ etc.
Under these isomorphism $j\_\*$ corresponds to the dual to restriction $j^\*$. This is $G$-equivariant by functoriality.
I'll let you check that $(g^\*\alpha,g^\*\beta)=(\alpha,\beta)$, and this implies $G$-equivariance of the duality isomorphisms, and that's all you need.


