Rationally connected Galois covers of Fano-Mori fibre spaces

For Fano-Mori fibre spaces $\pi \text{\hspace{0.17em}⁣}:V\to S$, every fibre of which is a primitive Fano variety with the global canonical threshold at least 1, and which are stable with respect to birational modifications of the base and sufficiently twisted over the base, we prove that every rationally connected Galois cyclic rational cover is pulled back from the base; in particular, if $S$ does not have such covers then $V$ does not have rationally connected Galois rational covers, the Galois group of which is not perfect.