Flatness and Cohen-Macaulay rings

Question

Let $A$ be a local Artin ring, $R$ a local Noetherian ring, $f:A \to R$ a flat morphism and $R$ is cohen-Macaulay. Let $I$ be an ideal in $R$ such that $R/I$ is also Cohen-Macaulay. Under what condition on $R/I$ can we conclude that $R/I$ is flat over $A$ i.e., the composition $A \to R \to R/I$ is flat? Recall that the answer to this question is well-known in the case $A$ is regular.