  Let $\mathcal{F}(h)$ be the number of imaginary quadratic fields with class
number $h$. In this note, we improve the error term in Soundararajan's
asymptotic formula for the average of $\mathcal{F}(h)$. Our argument leads to a
similar refinement of the asymptotic for the average of $\mathcal{F}(h)$ over
odd $h$, which was recently obtained by Holmin, Jones, Kurlberg, McLeman and
Petersen.
