Linear maps preserving invariants

Let $G\subset\mathrm{GL}(V)$ be a complex reductive group. Let $G'$ denote $\{\varphi\in\mathrm{GL}(V)\mid p\circ\varphi=p$ for all $p\in\mathbb{C}[V]^G\}$. We show that, ``in general'', $G'=G$. In case $G$ is the adjoint group of a simple Lie algebra $\mathfrak{g}$, we show that $G'$ is an order 2 extension of $G$. We also calculate $G'$ for all representations of $\mathrm{SL}_2$.