  We study the projective normality of a minimal surface $X$ which is a
ramified double covering over a rational surface $S$ with $\dim|-K_S|\ge 1$. In
particular Horikawa surfaces, the minimal surfaces of general type with
$K^2_X=2p_g(X)-4$, are of this type, up to resolution of singularities. Let
$\pi$ be the covering map from $X$ to $S$. We show that the
$\mathbb{Z}_2$-invariant adjoint divisors $K_X+r\pi^*A$ are normally generated,
where the integer $r\ge 3$ and $A$ is an ample divisor on $S$.
