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.