Let S be a minimal surface of general type with $p_g(S)=q(S)=0$ having an involution ${sigma}$ over the field of complex numbers. It is well known that if the bicanonical map ${varphi}$ of S is composed with ${sigma}$, then the minimal resolution W of the quotient $S/{sigma}$ is rational or birational to an Enriques surface. In this paper we prove that the surface W of S with $K^2_S=5,6,7,8$ having an involution ${sigma}$ with which the bicanonical map ${varphi}$ of S is composed is rational. This result applies in part to surfaces S with $K^2_S=5$ for which ${varphi}$ has degree 4 and is composed with an involution ${sigma}$. Also we list the examples available in the literature for the given $K^2_S$ and the degree of ${varphi}$.