  For an almost product structure $J$ on a manifold $M$ of dimension $6$ with
non-degenerate Nijenhuis tensor $N_J$, we show that the automorphism group
$G=Aut(M,J)$ has dimension at most 14. In the case of equality $G$ is the
exceptional Lie group $G_2^*$. The next possible symmetry dimension is proved
to be equal to 10, and $G$ has Lie algebra $sp(4,R)$. Both maximal and
submaximal symmetric structures are globally homogeneous and strictly nearly
para-K\"ahler. We also demonstrate that whenever the symmetry dimension is at
least 9, then the automorphism algebra acts locally transitively.
