On finite soluble groups with almost fixed-point-free automorphisms of noncoprime order

It is proved that if a finite p-soluble group G admits an automorphism φ of order pn having at most m fixed points on every φ-invariant elementary abelian p′-section of G, then the p-length of G is bounded above in terms of pn and m; if in addition G is soluble, then the Fitting height of G is bounded above in terms of pn and m. It is also proved that if a finite soluble group G admits an automorphism ψ of order paqb for some primes p and q, then the Fitting height of G is bounded above in terms of |ψ| and |CG(ψ)|.