  Let $G$ be a simple algebraic group of adjoint type over the field
$\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ 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 $G$-twisted
variety. In this article, we compute the connected component containing the
identity automorphism of the group of all algebraic automorphisms of some
particular $G$-twisted variety $E.$
