I am looking for references that may formalize the following idea: let $R = k[X]$ be the coordinate ring for a generic $n \times m$ matrix $M$. It is well known that the ideal of $r \times r$ minors is invariant under the induced $GL_n \times GL_m$-action. Suppose instead that one has $2$ different groups $G$ and $H$ acting on the variables of $R$, and the ideals $I$ and $J$ that are invariant under the action by $G$ and $H$, respectively. Then $I + J$ is acted on by both $G$ and $H$, but it is not acted on by $G \times H$, mainly because the actions by $G$ and $H$ do not necessarily commute.
My question is the following: is it still possible to view $I+J$ as having an action by some other group, involving $G$ and $H$? Or, have objects with "partial" group actions (like $I+J$ as above) been studied before?