A Fitting class \(\mathcal F\) is called dominant in the class \(\mathcal S\) of all finite soluble groups if \(\mathcal F\subseteq\mathcal S\) and for every group \(G\in\mathcal S\) any two \(\mathcal F\)-maximal subgroups of \(G\) containing the \(\mathcal F\)-radical \(G_{\mathcal F}\) are conjugated in \(G\). Lockett in 1971 proved that if \(\mathcal F\) is a soluble dominant Fitting class, then either the class \(\mathcal N\) of finite nilpotent groups is contained in \(\mathcal F\) or \(\mathcal F=\mathcal S_\pi\). A Fitting class \(\mathcal F\) is called local if \(\mathcal F=\mathcal S_\pi\cap\bigcap_{p\in\pi}f(p)\mathcal S_p\mathcal S_{p'}\), where \(f\) is a map which assigns to each prime \(p\) a Fitting class \(f(p)\) and \(\pi=\text{Char\,}\mathcal F\). Local Fitting classes constitute a large family of Fitting classes, though not every Fitting class is local and not every local Fitting class is dominant in \(\mathcal S\). An interesting problem is to decide which local Fitting classes are dominant. The main result in this paper settles this question and gives a complete characterization of dominant local Fitting classes containing \(\mathcal N\).