On $3$-designs from $PGL(2,q)$

The group $PGL(2,q)$ acts $3$-transitively on the projective line $GF(q) \cup \{\infty\}$. Thus, an orbit of its action on the $k$-subsets of the projective line is the block set of a $3$-$(q+1,k,\lambda)$ design. We find the parameters of the designs formed by the orbit of a block of the form $\langle \theta^r \rangle$ or $\langle \theta^r \rangle \cup \{ 0\}$, where $\theta$ is a primitive element of $GF(q)$.