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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 groupof 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 threevariables fixing one variable) can be mimicked by tame automorphisms. This onthe one hand removes the hope of showing that such an automorphism is non-tameby studying one bijection it induces, but on the other hand indicates that thebijections 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 bemimicked by the Derksen subgroup $\DA_n(\F_q)$, which is the subgroup generatedby affine maps and one particular element. It is known that if a field $k$ hascharacteristic zero, then $\TA_n(k)=\DA_n(k)$, which is not expected to be truefor characteristic $p$.  In section 6 we consider the subgroups $\GLIN_n(k)$ and $\GTAM_n(k)$ of thepolynomial automorphism group (which are candidates to equal the entireautomorphism group). We show that $\GLIN_n(\F_q)=\GTAM_n(\F_q)$ if $q\not = 2$,and $\GLIN_n(\F_2)\subsetneq \GTAM_n(\F_2)$.

Polynomial automorphisms over finite fields: Mimicking non-tame and tame maps by the Derksen group