Isomorphisms between $\mathrm{Aut}$ of vector spaces.

Question

If we have an isomorphism $\phi: V \to W$ of $\mathbb{F}$-vector spaces, $\mathbb{F}$ a field, this induces an isomorphism $\hat \phi: \mathrm{Aut}(V) \to \mathrm{Aut}(W)$ by "conjugation" ($\hat \phi(\psi):=\phi\psi\phi^{-1}$ for every $\psi \in \mathrm{Aut}(V)$). Is every isomorphism $\mathrm{Aut}(V) \to \mathrm{Aut}(W)$ of this form?