  We completely determine the localized automorphisms of the Cuntz algebras
$O_n$ corresponding to permutation matrices in $M_n \otimes M_n$ for $n=3$ and
$n=4$. This result is obtained through a combination of general combinatorial
techniques and large scale computer calculations. Our analysis proceeds
according to the general scheme proposed in a previous paper, where we analyzed
in detail the case of $O_2$ using labeled rooted trees. We also discuss those
proper endomorphisms of these Cuntz algebras which restrict to automorphisms of
their respective diagonals. In the case of $O_3$ we compute the number of
automorphisms of the diagonal induced by permutation matrices in $M_3 \otimes
M_3 \otimes M_3$.
