  If $F$ is a polynomial automorphism over a finite field $\F_q$ in dimension
$n$, then it induces a bijection $\pi_{q^r}(F)$ of $(\F_{q^r})^n$ for every
$r\in \N^*$. We say that $F$ can be `mimicked' by elements of a certain group
of automorphisms $\mathcal{G}$ if there are $g_r\in \mathcal{G}$ such that
$\pi_{q^r}(g_r)=\pi_{q^r}(F)$.
  We show that the Nagata automorphism (and any other automorphism in three
variables fixing one variable) can be mimicked by tame automorphisms. This on
the one hand removes the hope of showing that such an automorphism is non-tame
by studying one bijection it induces, but on the other hand indicates that the
bijections of tame automorphisms coincide with bijections of all automorphisms.
  In section 5 we show that the whole tame group $\TA_n(\F_q)$ in turn can be
mimicked by the Derksen subgroup $\DA_n(\F_q)$, which is the subgroup generated
by affine maps and one particular element. It is known that if a field $k$ has
characteristic zero, then $\TA_n(k)=\DA_n(k)$, which is not expected to be true
for characteristic $p$.
  In section 6 we consider the subgroups $\GLIN_n(k)$ and $\GTAM_n(k)$ of the
polynomial automorphism group (which are candidates to equal the entire
automorphism group). We show that $\GLIN_n(\F_q)=\GTAM_n(\F_q)$ if $q\not = 2$,
and $\GLIN_n(\F_2)\subsetneq \GTAM_n(\F_2)$.
