Almost simple groups of Suzuki type acting on polytopes

Let $S = Sz(q)$, with $q\neq 2$ an odd power of two. For each almost simple group $G$ such that $S < G \leq Aut(S)$, we prove that $G$ is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For $G = Sz(q)$, we show that there is always at least one abstract regular polytope $\mathcal{P}$ such that $G = Aut(\mathcal{P})$. Moreover, if $\mathcal{P}$ is an abstract regular polytope such that $G = Aut(\mathcal{P})$, then $\mathcal{P}$ is a polyhedron.