Let $X$ be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and non-positive signature $\sigma(X)$. In this paper we use Seiberg-Witten theory to prove that if $X$ admits a spin alternating $A_5$ action, then $b_2^+(X)\geq |\sigma(X)|/8 +3$ under some non-degeneracy conditions.
