On the irreducibility of symmetrizations of cross-characteristic representations of finite classical groups

Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\mathrm{\ne }\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module $W$, chosen minimally with respect to containing the image of $H$ under the associated representation. We consider the question of when $H$ can act irreducibly on a $G$-constituent of $W^{\otimes e}$ and study its relationship to the maximal subgroup problem for finite classical groups.