  We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an
involution $i$ such that the bicanonical map of $S$ is not composed with $i$.
It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an
Enriques surface or it has Kodaira dimension $1$ and the possibilities for the
ramification divisor of the covering map $S\rightarrow S/i$ are described. We
also show that these two cases do occur, providing an example. In this example
$S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$
is of degree $2$ onto a rational surface.
