Abstract : 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 $K_{S}^{2}=5,6,7,8$ 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 $K_{S}^{2}=5$ 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 $K_{S}^{2}$ and the degree of $\varphi$.
