I would like to mention a few topics regarding symmetries of finite graphs that I find interesting, intriguing and worth studying.
The first topic is about lifting automorphisms along covering projections. Suppose one is given a finite connected graph $\Gamma$ and a group of automorphisms $G$ acting on it. Can one find a regular covering projection $\wp$ onto $\Gamma$ such that $G$ is the maximal group that lifts along $\wp$ and such that the full automorphism group of the graph is the lift of $G$? A recent partial result answer proved recently my Pablo Spiga and myself will be presented.
The second topic is about vertex-transitive graphs admitting an automorphism fixing many vertices; here a strict definition of the term ``many'' is intentionally avoided so that by varying it one can prove different results. Some computational data regarding cubic vertex-transitive graphs will be presented.
If time permits, a third topic regarding vertex-transitive graphs admitting an automorphism with a long orbit will be discussed; here the term ``long'' means a suitable fixed proportion of the order of the graph. Some results obtained recently by Micael Toledo about the cubic case will be mentioned.