Automorphisms of the ring of invariants of the binary quintic representation of SL2

Let k⁶ denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2 (k) on k⁶ induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k⁶) SL2 of invariant polynomials and show that Autₖ (Q) = u (k), the unit group of k, where Autₖ (Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k⁶ is isomorphic to u (k).