the relation between automorphism group and isometry group of left invariant (α, β)-metrics

Let f be an ( alpha;, beta;)-metric which is defined by a left invariant vector field and a left invariant riemannian metric on a simply connected real lie group g. we consider the automorphism and isometry groups of the finsler manifold (g,f) and their intersection. we prove that for an arbitrary left invariant vector field x and any compact subgroup k of automorphisms which x is invariant under them, there exists an ( alpha;, beta;)-metric such that k is a subgroup of its isometry group.