  In this paper, we classify the minimal surfaces of general type with
$\chi=5$, $K^{2}=9$ whose canonical map is composed with an involution. We
obtain 6 families, whose dimensions in the moduli space are 28, 27, 33, 32, 31
and 32 respectively. Among them, the family of surfaces having a genus 2
fibration forms an irreducible component of $\mathfrak{M}_{\chi=5, K^{2}=9}$.
