  A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also
called the canonical double cover of $X$) has automorphisms that do not come
from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if
it is unstable, connected, and non-bipartite, and no two distinct vertices of X
have exactly the same neighbors.
  We find all of the nontrivially unstable circulant graphs of valency at most
$7$. (They come in several infinite families.) We also show that the
instability of each of these graphs is explained by theorems of Steve Wilson.
This is best possible, because there is a nontrivially unstable circulant graph
of valency $8$ that does not satisfy the hypotheses of any of Wilson's four
instability theorems for circulant graphs.
