Let $A$ be an abelian variety of dimension $g$ over $C$ with complex multiplication by a CM field $K$ where $[K:Q] =2g$. By this I mean that End($A$) $\cong \mathcal{O}_K$. Then, $A$ has a model over the Hilbert class field of the reflex field of $K$ which I denote by $H$. Let $\mathfrak{P}$ be a prime of $H$ above a rational prime which splits completely in the reflex field of $K$. Furthermore, assume that $A$ has good reduction at this prime.

My question is does the reduction of $A$ modulo $\mathfrak{P}$ have complex multiplication by $K$? If not, are there some splitting conditions one can put on the prime $\mathfrak{P}$ to ensure that the reduction has complex multiplication?