  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 $Z_2$.
