Let \(\Omega\) be a finite group of operators and \(\mathcal M\) a variety of \(\Omega\)-groups, defined by the \(\Omega\)-identities \(\{v_ \alpha\}\). Then \(\{\overline{v_ \alpha}\}\) denotes the family of (ordinary) group identities which is obtained from \(\{v_ \alpha\}\) by replacing all operators from \(\Omega\) by \(1\) and \(\overline{\mathcal M}\) denotes the variety of groups defined by \(\{\overline{v_ \alpha}\}\). Suppose that locally nilpotent groups in \(\overline{\mathcal M}\) constitute a subvariety and that the associated Lie ring of a free group of \(\overline{\mathcal M}\) satisfies a system of multilinear identities that define a locally nilpotent variety of Lie rings in which the nilpotency class of a \(d\)- generator ring is bounded by a function \(f(d)\). Then it is proved in this paper that if a group \(G \in {\mathcal M}\) is such that the semidirect product of \(G\) and \(\Omega\) is locally nilpotent, then \(G\) belongs to a locally nilpotent variety in which the nilpotency class of a \(d\)- generator group is bounded by the function \(f(d \cdot (| \Omega|^{| \Omega|} - 1)/(| \Omega| -1))\). Some applications of the theorem are given for the case in which \(\Omega\) is a finite \(p\)-group. An example shows that the condition that the Lie ring identities be multilinear is necessary.