Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $\varphi$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $\varphi$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $\varphi$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polynomial and $G$ is a $\sigma'$-group for a finite set of primes $\sigma=\sigma(f(x))$ depending only on $f(x)$, then the Fitting height of $G$ is bounded in terms of the number $\operatorname{irr}(f(x))$ of irreducible factors in the decomposition of $f(x)$. These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length $\alpha (|\varphi|)$ of $\langle\varphi\rangle$ when $\operatorname{deg} (f(x))$ or $\operatorname{irr}(f(x))$ is small in comparison with $\alpha (|\varphi|)$.