On the Fitting length of generalized Hughes subgroup (Q1121381)

In the search of reasonable generalizations of a result of \textit{A. Espuelas} [J. Algebra 105, 365-371 (1987; Zbl 0604.20021)] the following result is obtained. Theorem: Let H be a finite, solvable group admitting an automorphism \(\alpha\) of prime order p. Suppose that \([H,\alpha]=H\) and let \(G=H<\alpha >\) be the semidirect product of H and \(<\alpha >\). If the order of every element in \(G\setminus H\) divides p.q where q is a prime different from p, then the Fitting length of H is at most 2.