  Let $F$ be a nilpotent group acted on by a group $H$ via automorphisms and
let the group $G$ admit the semidirect product $FH$ as a group of automorphisms
so that $C_G(F) = 1$. We prove that the order of $\gamma_\infty(G)$, the rank
of $\gamma_\infty(G)$ are bounded in terms of the orders of
$\gamma_{\infty}(C_G(H))$ and $H$, the rank of $\gamma_{\infty}(C_G(H))$ and
the order of $H$, respectively in cases where either $FH$ is a Frobenius group;
$FH$ is a Frobenius-like group satisfying some certain conditions; or
$FH=\langle \alpha,\beta\rangle$ is a dihedral group generated by the
involutions $\alpha$ and $\beta$ with $F =\langle \alpha\beta\rangle$ and $H
=\langle\alpha \rangle$.
