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In $\mathrm{PG}(3, q)$, $q = 2^n$, $n \ge 3$, let ${\cal A} =\{(1,t,t^{2^h},t^{2^h+1}) \mid t \in \mathbb{F}_q\} \cup \{(0,0,0,1)\}$, with$\mathrm{gcd}(n,h) = 1$, be a $(q+1)$-arc and let $G_h \simeq \mathrm{PGL}(2,q)$ be the stabilizer of $\cal A$ in $\mathrm{PGL}(4, q)$. The $G_h$-orbits onpoints, lines and planes of $\mathrm{PG}(3, q)$, together with the point-planeincidence matrix with respect to the $G_h$-orbits on points and planes of$\mathrm{PG}(3, q)$ are determined. The point-line incidence matrix withrespect to the $G_1$-orbits on points and lines of $\mathrm{PG}(3, q)$ is alsoconsidered. In particular, for a line $\ell$ belonging to a given line$G_1$-orbits, say $\cal L$, the point $G_1$-orbit distribution of $\ell$ iseither explicitly computed or it is shown to depend on the number of elements$x$ in $\mathbb{F}_q$ (or in a subset of $\mathbb{F}_q$) such that$\mathrm{Tr}_{q|2}(g(x)) = 0$, where $g$ is an $\mathbb{F}_q$-map determined by$\cal L$.

On the geometry of a $(q + 1)$-arc of $\mathrm{PG}(3, q)$, q even