3-Designs from PSL(2,q) with cyclic starter blocks

We consider when the projective special linear group over a finite field defines a $3$-design with a cyclic starter block. We will show that the equivalences of the existence of such $3$-$(q+1,5,3)$ and $3$-$(q+1,10,18)$ designs for a prime power $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}20)$, and $3$-$(q+1,13,33)$ and $3$-$(q+1,26,150)$ designs for a prime power $q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}52)$, respectively.