Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.

Associated to $\chi$ is a projection from $W$ onto a subspace, which we will denote by $\pi_\chi$ (this is standard, but I can write down the formula for $\pi_\chi$ if needed). We will also denote by $\chi$ the element of the group algebra $\mathbb{C}[S_n]$ corresponding to $\pi_\chi$. Let us now form $$ \rho = \sum_{h \in H} h \chi h^{-1} \in \mathbb{C}[S_n].$$ Note that, for any $h \in H$, we have $h \rho = \rho h$.

I am interested in the subspace of $W$ obtained as the image of $\rho$. How can one decompose it into irreducibles, under the action of $\operatorname{SL}(2, \mathbb{C})$? Has it been studied in the literature?

Edit 1: it turns out that actually, the $\rho$ I am interested in is of the form $$ \rho = \sum_{h \in H} h \chi \in \mathbb{C}[S_n],$$ so there is no conjugation, unlike what I had previously thought. @Will Sawin suggested that I apply the Schur-Weyl duality, which is indeed the right tool to use for this question.