Reduced mod $p$ homology of a $p$-complete Eilenberg-MacLane space

Question

Let $A$ be a $p$-complete abelian group for some prime $p$. Is it true that $\tilde{H}_*(K(A,2);\mathbb{F}_p)=0$? If so, how can one prove it?

Please, also let me know if this hold only if we require other properties from $A$!