Thompson proved that a finite group with a fixed point free automorphism of prime order is nilpotent. Kegel proved the same result for splitting automorphisms of prime order. Prob- lems about splitting automorphisms of finite solvable groups are studied by Jabara, Kurzweil, Meixner and Khukhro. In this talk we will prove the following results: Theorem. (E.) A finite group with a splitting automorphism of odd order is solvable. Theorem:(E. - Gupta) For any $n\in N$, a finite group with a splitting automorphism of order $2^n$ is solvable.