Automorphisms in spaces of continuous functions on Valdivia compacta

We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(Γ). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(Γ), all isomorphism between subspaces of C(K) of size less than ℵω extend to automorphisms of C(K).