Let $G$, $K$ and $H$ be three finite groups where $H$ acts on both of $G$ and $K$ as sets nontrivially. Moreover assume that $\mathbb{C}K$ is (isomorphic to) a $\mathbb{C}H$-submodule of $\mathbb{C}G$ (where $\mathbb{C}$ is field of complex numbers).
Is it true that $K$ must be (isomorphic to) a $H$-subgroup of $G$?
The group structures of $G$ and $K$ don't affect the hypotheses, so you shouldn't expect to be able to deduce much about them.
For any groups with $|K|<|G|$ and any action of $H$ on $K$, you could just identify $K$ (as a set) with some subset $K'$ of $G$, and let $H$ act on $G$ by acting the same way on $K'$ as it does on $K$, and in an arbitrary way on $G\setminus K'$. Then $\mathbb{C}G\cong\mathbb{C}K\oplus\mathbb{C}[G\setminus K']$ as $\mathbb{C}H$-modules, so $\mathbb{C}K$ is ismomorphic to a $\mathbb{C}H$-submodule of $\mathbb{C}G$.