Let N/M be an inclusion of von Neumann algebras with a conditional expectation E: M Γ N satisfying the finite index condition of [PiPo], i.e., there exists c>0 such that E(x) cx, \x # M + . In [Po4] we proved that such inclusions N/M satisfy the relative version of Dixmier's property, namely for any x # M, the norm closure of the convex hull of the averaging elements of x by unitaries in N, C N (x)=co n [uxu* | u unitary element in N], contains elements from the relative commutant of N in M : C N (x) & N$ & M{<. The proof used at a key point the classical result of Dixmier for the single von Neumann algebra N ([D]), showing that for x # N the above averaging'' sets C N (x) satisfy C N (x) & Z(N){<, where Z(N)= N$ & N is the center of N. In this paper we investigate the C\*-algebra version of this result, proving the relative Dixmier property for certain inclusions of C\*-algebras B/A, with conditional expectations E: A Γ B satisfying the finite index condition in [PiPo]. As in the von Neumann algebra case treated in [Po4], the proof will depend on the validity of the Dixmier property for the single C\*-algebra B. Thus, our result roughly shows that the relative Dixmier property for an inclusion B/A with Ind(B/A)< holds whenever the single C\*-algebra B has itself the Dixmier property. Since the notion of center'' of an algebra doesn't fit so well in the context of C*-algebras (although it can be formally defined in the same way as for von Neumann algebras), for single C*-algebras B one takes the property C B (x) & C1{<, \x # B'' to be, by definition, the Dixmier property''.