It is known that linear groups are not closed under extensions, but what if the extension splits, i.e. it is a semidirect product?
Suppose that $K,R$ are subgroups of $\mathop{GL}(n,\mathbb{F})$, where $\mathbb{F}$ is a field, and suppose that I have a homomorphism $\phi \colon R \to \mathop{Aut}(K)$ which defines the semidirect product $G = K \rtimes_{\phi}R$. My question is, does $G$ embed into $\mathop{GL}(m,\mathbb{F})$ for some $m > n$? Or perhaps into $\mathop{GL}(m,\mathbb{F}')$, where $\mathbb{F}'$ is some extension of $\mathbb{F}$?