Transitive Automorphism Groups of Steiner Systems

Question

I recently did a project where I constructed the Mathieu groups $M_{11}$ and $M_{12}$ as automorphism groups from resp S(4,5,11) and S(5,6,12). I know that $M_{12}$ is sharply 5-transitive and $M_{11}$ sharply 4-transitive. Because $M_{24}$ is 5-transitive but not sharply 5-transitive I thought it might be the case that the automorphism group of a Steiner system S(t,k,v) is t-transitive or sharply t-transitive if k = t+1. I didn't come across such a theorem but I was wondering if it was true and how one would prove it?