  Minimal algebraic surfaces of general type $X$ such that
$K^2_X=2\chi(\mathcal{O}_X)-6$ are called Horikawa surfaces. In this note
$\mathbb{Z}_3$-actions on Horikawa surfaces are studied.
  The main result states that given an admissible pair $(K^2, \chi)$ such that
$K^2=2\chi-6$, Gieseker's moduli space $\mathfrak{M}_{K^2,\chi}$ contains
surfaces admitting a $\mathbb{Z}_3$-action. In addition, the connected
components of $\mathfrak{M}_{K^2,\chi}$ that do not contain such surfaces are
characterized.
  On the other hand, the examples considered allow to construct stable surfaces
that do not admit $\mathbb{Q}$-Gorenstein smoothings. This is illustrated by
constructing non-smoothable surfaces in the KSBA-compactification
$\overline{\mathfrak{M}}_{K^2,\chi}$ of Giesker's moduli space
$\mathfrak{M}_{K^2,\chi}$ for every admissible pair $(K^2, \chi)$ such that
$K^2=2\chi-5$.
