Suppose $\Gamma$ is a $k$regular graph with $n$vertex. What is the group structure of Automorphism of $\Gamma$?
Under these weak hypotheses, the answer is: it could be anything. The trivial group is possible, as well as $\mathfrak S_{n}$ and essentially any group in between by Frucht's theorem (which realizes any group as the automorphism group of a regular graph). Of course, there are trivial obstructions on $k$ and $n$ for some groups to appear (for instance $\mathfrak S_{n}$ of course appears only if $k=0$ or $k=n1$) but I don't see we can say much more beyond trivialities in the generality you are considering. 


There is a theorem of Wormald, a divisibility condition on the order of the group. See http://www.jstor.org/discover/10.2307/2043017?sid=21105736858423&uid=2129&uid=70&uid=4&uid=2&uid=3739256&uid=3739808 

