  A Howe curve is defined as the normalization of the fiber product over a
projective line of two hyperelliptic curves. Howe curves are very useful to
produce important classes of curves over fields of positive characteristic,
e.g., maximal, superspecial, or supersingular ones. Determining their feasible
equations explicitly is a basic problem, and it has been solved in the
hyperelliptic case and in the non-hyperelliptic case with genus not greater
than $4$. In this paper, we construct an explicit plane sextic model for
non-hyperelliptic Howe curves of genus $5$. We also determine the number and
type of singularities on our sextic model, and prove that the singularities are
generically $4$ double points. Our results together with Moriya-Kudo's recent
ones imply that for each $s \in \{2,3,4,5\}$, there exists a non-hyperellptic
curve $H$ of genus $5$ with $\mathrm{Aut}(H) \supset \mathbf{V}_4$ such that
its associated plane sextic has $s$ double points.
