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In [R] explicit representatives for $S^3$-principal bundles over $S^7$ are constructed, based on these constructions explicit free $S^3$-actions on the total spaces are described, with quotients exotic $7$-spheres. To describe these actions a classification formula for the bundles is used. This formula is not correct. In Theorem 1 below, we correct the classification formula and in Theorem 2 we exhibit the correct indices of the exotic $7$-spheres that occur as quotients of the free $S^3$-actions described above.

Let p be a prime number and X a simply connected Hausdorff space equipped with a free ${\mathbb{Z}}_p$-action generated by $f_p:X\to X$. Let $\alpha :S^{2n-1}\to S^{2n-1}$ be a homeomorphism generating a free ${\mathbb{Z}}_p$-action on the (2n-1)-sphere, whose orbit space is some lens space. We prove that, under some homotopy conditions on X, there exists an equivariant map $F:(S^{2n-1},\alpha )\to (X,f_p)$. As applications, we derive new versions of generalized Lusternik-Schnirelmann and Borsuk-Ulam theorems.