  We determine the number of Del Pezzo surfaces of degree 2 over finite fields
of odd characteristic with specified action of the Frobenius endomorphism, i.e.
we solve the "quantitative inverse Galois problem". As applications we
determine the number of Del Pezzo surfaces of degree 2 with a given number of
points and recover results of Banwait-Fit\'e-Loughran and Loughran-Trepalin.
