Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

It is known that a finite group with an automorphism \(\varphi\) of coprime order has a soluble radical of \((|\varphi|,|C_G(\varphi)|)\)-bounded Fitting height and index. We extend this classic result as follows. Let \(f(x) = a_0 + a_1 \cdot x + \cdots + a_d \cdot x^d \in \mathbb{Z}[x]\) be a primitive polynomial and let \(G\) be a finite group with an automorphism \(\varphi\) of coprime order satisfying \( g^{a_0} \cdot \varphi(g)^{a_1} \cdots \varphi^d(g)^{a_d} = 1 \), for all \(g \in G\). Then the soluble radical of \(G\) has \((d,|C_G(\varphi)|)\)-boundex Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree \(d\).