2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73355We show that for any prime p and for any II$_1$ factor N there exist two $mathbb{Z}_{p^2}$-- actions on the free product factor $*_{1} ^{p}N$ that have the same outer invariant but are not outer conjugate. Therefore, in the case of free product factors, the outer invariant is not a complete invariant for outer conjugacy.32 pages, corrected typos in title and abstractOperator Algebras46L54, 46L40, 46L37text