Let X be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and signature ${sigma}-(X)$. In this paper we use Seiberg-Witten theory to prove that if X admits a spin alternating $A_4$ action, then $b^+_2(X)$ ${geq}$ |${sigma}{(X)}$|/8+3 under some non-degeneracy conditions.