It is known that an abelian variety over a finite field may not admit a lifting to an abelian variety with complex multiplication in characteristic 0. The question of strong CM lifting (sCML) asks whether we can kill the obstructions to a CM lifting by requiring the whole ring of integers in the CM field act on the abelian variety. I will give counterexamples to question (sCML), and prove the answer to question (sCML) is affirmative under the following assumptions on the CM field L: for every place v above p in the maximal totally real subfield L_0 , either v is inert in L, or v is split in L with absolute ramification index e(v) < p − 1.