On Automorphism group of a $G$-induced variety

Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G\mathrm{.}$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety $E:=G{\times }^{B}F,$ which has a natural action of $G$; we call it $G$-induced variety or $(G,B)$-induced variety. In this article, we compute the connected component containing the identity automorphism of the group of all algebraic automorphisms of some particular $G$-induced varieties $E\mathrm{.}$