permutation polynomial

Question

If we have $\operatorname{GF}(4)$ as an extension field, we can define a permutation polynomial of $\operatorname{GF}(4)$ like $L(x)$, a linearized polynomial, of the followinf form: $L(x)= \sum\limits_{s=0}^{r-1} a_s x^{q^s} $ Is it possible to get more details for $\operatorname{GF}(4)$ and $\operatorname{GF}(2)$. I want to know more about Betti-Mathieu group. What are the permutation polynomials of $\operatorname{GF}(4)$ over $\operatorname{GF}(2)$? According to mentioned group the set of $L(x)$ is isomorph to $\operatorname{GL}(2,2)$ in my case. Actually I need to know more about this notion with more practical examples. How many permutation polynomial there exist for $\operatorname{GF}(4)$ over $\operatorname{GF}(2)$?