A free G-group is a non-abelian analogue of a free G-module; it is a free group with G acting freely on a set of free generators. The equivariant automorphism group of the free G-group of rank n occurred in Kiralis's study of ``non-abelian K-theory'' [K]. It was independently introduced also by Denk-Metzler [DM] and McCool [M] who gave an infinite presentation of it. The paper  contains an extensive study of the natural homomorphism , and it brings together ideas of combinatorial group theory and K-theory. For example, it is known that the kernel of is the normal closure of a single element; we prove that if is the quotient of by the normal closure of the same element, then the kernel and the cokernel of the homomorphism are the unstable Whitehead groups Wh2,n(G) and Wh1,n(G) respectively. We also prove that if G is finitely presented and , then many of the subgroups we study, including , are finitely presented. The case n=2 is exceptional, and in  we prove that is not finitely presentable whenever the abelianization G/[G,G] is infinite.