If $G$ is a classical semisimple Lie group acting on a vector $\mathbb{C}$-space $V$ then it is well known that the algebra of invariants $\mathbb{C}[V]^G$ councides with the algebra of invariants $\mathbb{C}[V]^L$ of corresponding Lie algebra $L.$

Question. Let now $G$ be a finite group, $V$ be its representation and $\mathbb{C}[V]^G$. Is there existі a Lie algebra $L$ and its representation on $V$ such that $\mathbb{C}[V]^G=\mathbb{C}[V]^L$?

It is easy to see that it impossible for symmetric group $S_n$. But maybe there are classes of finite groups for which can be found the positive answer?