Let alpha be an automorphism of a hyperelliptic curve C of genus g,
and let alpha' be the automorphism of P^1 induced by alpha.
Let n be the order of alpha and let n' be the order of alpha'.
We show that the triple (g,n,n') completely determines the
characteristic polynomial of the automorphism alpha^* of the
Jacobian of C, unless n is even, n=n', and (2g+2)/n is even,
in which case there are two possibilities. We give explicit
formulas for the characteristic polynomial in all cases.