Let $G$ be a complex reductive group, $X$ a smooth projective variety on which $G$ acts algebraically, and $Y \subseteq X$ a $G$-invariant smooth closed subvariety such that $X\setminus Y$ is also smooth.

If $d$ is twice the complex codimension of $Y$ in $X$, then my understanding of the equivariant Gysin sequence (with coefficients in $\mathbb Q$) is that there is a $\mathbb Q$-module isomorphism $$ H_G^{*}(X, X\setminus Y) \to H_G^{*-d}(Y)$$ which turns the long exact sequence for relative cohomology into the Gysin sequence $$ \rightarrow H_G^{*-d}(Y) \xrightarrow{g} H_G^*(X) \xrightarrow h H_G^*(X\setminus Y) \rightarrow.$$ Of interest to me, is whether the map $g: H_G^{*-d}(Y) \rightarrow H_G^*(X)$ is an $H_G^*(\text{pt})$-module morphism.

As $h:H_G^*(X) \rightarrow H_G^*(X\setminus Y)$ is the dual of an inclusion, it is an $H_G^*(\text{pt})$-algebra morphism, but I cannot seem to work out any details about $g$. In particular, I feel that it is likely not an algebra map, but if anyone has any insight into whether it is a module map it would be greatly appreciated.