Introduction. Paper [1] was a natural continuation of investigation of basic problems of the theory of geodesic mappings of pseudo-Riemannian spaces. This theory was originated in the end of the 19th century by T. Levi-Civita, T. Tomas, and H. Weyl. Since then many mathematicians worked in this area of research, studying geodesic mappings of pseudo-Riemannian manifolds endowed with additional structures. The theory was enriched by new results. Thus, in the middle of the 20th century, Westlake and Yano proved that K ¨ ahler manifolds admit no nontrivial geodesic transformations preserving the complex structure. Recently a contact analog of these results has been obtained. In particular, in [1] (p. 215), the notion of a contact-geodesic transformation of an almost contact metric structure was introduced as a geodesic transformation preserving the almost contact structure. In the same paper, a technique was developed with the use of which it was proved that cosymplectic and Sasakian structures, as well as Kenmotsu structures, admit no nontrivial contact-geodesic transformations of the metric. In this paper, using the results of the above mentioned research, we prove that quasi-Sasakian structures admit no nontrivial contact-geodesic transformations of the metric, which, in turn, generalizes results of [1]. We also prove that regular locally conformally quasi-Sasakian structures admitting nontrivial contact-geodesic transformations of the metric are normal.