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Let G be a finite group and A be a subgroup of Aut(G). In this work, we studied the influence of the index of fixed point subgroup of A in G on the structure of G. When A is cyclic, we proved the following: (1) [G,A] is solvable if this index is squarefree and the orders of G and A are coprime. (2) G is solvable if the index of the centralizer of each x in H-G is squarefree where H denotes the semidirect product of G by A. Moreover, for an arbitrary subgroup A of Aut(G) whose order is coprime to the order of G, we showed that when G is solvable, then the Fitting length f([G,A]) of [G,A] is bounded above by the number of primes (counted with multiplicities) dividing the index of fixed point subgroup of A in G and this bound is best possible.Ph.D. - Doctoral Progra