In Hartshorne's book "Deformation Theory" one can find a statement (inside the proof of Theorem 10.1) that every deformation $X' \to Spec(C)$ of an affine scheme $$X = Spec(k[x_1,\ldots,x_n]/(f_1,\ldots,f_i)) \to Spec(k)$$ for $C \to k$ is in fact isomorphic to an embedded deformation, i.e., it arises from lifting of the elements $f_j$ to $C[x_1,\ldots,x_n]$.

Is this also true for closed subschemes of regular local rings? Namely, if one fixes a regular local rings $R/k$ together with a flat $C$-lifting $R'/C$ of $R$. Does any flat lifting of $Spec(R/(f_1,\ldots,f_i))$ necessarily come from a lifting of $f_j$'s to $R'$? Maybe, there are some special cases concerning $R$ being essentially of finite type over some field $k$? Is there any additional condition one can impose on the sequence $(f_1,\ldots,f_i)$ (regular sequence? for example) to make such a statement true?