Infinitesimal Torelli theorem for double coverings of surfaces of general type

Let $X$ be a smooth complex projective surface subject to the following conditions:

1. [(i)] the canonical divisor $K_X$ of $X$ is ample,

2. [(ii)] the irregularity $q(X) = h^1(\mathcal {O}_X) =0$ and $p_g (X) =h^0 (\mathcal {O}_X (K_X)) \geq 2$,

3. [(iii)] the canonical linear system $\mid K_X\mid$ contains a reduced irreducible divisor.

It is shown that if $K^2_X \geq 5$, then the Infinitesimal Torelli theorem holds for a double covering of $X$ branched along a smooth divisor in the linear system $\mid 2K_X\mid$.