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We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having aninvolution $i$ 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 anEnriques surface or it has Kodaira dimension $1$ and the possibilities for theramification divisor of the covering map $S\rightarrow S/i$ are described. Wealso 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.

Involutions on surfaces with p g = q = 0 and K 2 = 3