Typical automorphism groups of finite nonrigid structures

We work with a finite relational vocabulary with at least one relation symbol with arity at least 2. Fix any integer $m > 1$. For almost all finite structures (labelled or unlabelled) such that at least $m$ elements are moved by some automorphisms, the automorphism group is $(Z_2)^i$ for some $i \leq (m+1)/2$; and if some relation symbol has arity at least 3, then the automorphism group is almost always $Z_2$.