Non-Outer Conjugate $\mathbb{Z}_{p^2}$ --Actions on Free Product Factors
  We 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. 
Kenneth Dykema and Maria Grazia Viola