Nonlinear permutations of a vector space recursively generated over a Galois ring of characteristic 4

For any integers $r\geq1$ and $m\geq3$, some class of nonlinear permutation of a vector space $(\operatorname{GF}(2^r))^m$ is constructed. Every permutation in the class is defined as a composition of two operations: (1) a linear recurring transformation with a characteristic polynomial $F(x)$ over a Galois ring $R$ of cardinality $2^{2r}$ and characteristic 4; and (2) taking the first digit in an element of $R$ represented by a pair of elements from $\operatorname{GF}(2^r)$. A necessary and sufficient condition is pointed for $F(x)$ of a certain type in the composition to provide the bijectiveness property of the composition.