For a finite group $G$ the regular action $\rho$ of $G$ on itself (by right multiplication) has the property that the normalizer of $\rho(G)$ in the symmetric group $S_G$ is isomorphic to the holomorph $G\rtimes \mathop{Aut}(G)\;$. [I got reminded to this fact by this recent question at mathoverflow.]
What is the according normalizer in $\mathop{GL}_F(V)$ when taking $G$ as a base of a vector space $V$ over a field $F$ (which can be chosen as you like) with $G$ acting regularly on this base?