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Let G be a finite group and A ≤ Aut(G). The index |G:CG(A)| is called the index of A in G and is denoted by IndG(A). In this paper, we study the influence of IndG(A) on the structure of G and prove that [G, A] is solvable in case where A is cyclic, IndG(A) is squarefree and the orders of G and A are coprime. Moreover, for arbitrary A ≤ Aut(G) whose order is coprime to the order of G, we show that when [G;A] is solvable, the Fitting height of [G,A] is bounded above by the number of primes (counted with multiplicities) dividing IndG(A) and this bound is best possible. [ABSTRACT FROM AUTHOR]