Proposition 1

If $\varphi$ admits a lifting, the mapping $(\delta, \tilde{\varphi}) \mapsto \delta + \tilde{\varphi}$ defines a simply transitive operation of the group of $k$-derivations of $A$ into $N$ on the set of liftings of $\varphi$.

Let $\tilde{\varphi}_0 : A \to C$ be a lifting of $\varphi$. The mapping $\delta \mapsto \delta + \tilde{\varphi}_0$ induces a bijection of the set of mappings of $A$ into $N$ onto the set of mappings $\tilde{\varphi} : A \to C$ such that $\pi \circ \tilde{\varphi} = \varphi$. Fix $\delta$, and put $\tilde{\varphi} = \delta + \tilde{\varphi}_0$. In order that $\tilde{\varphi}$ be a homomorphism of $k$-algebras, it is necessary and sufficient that $\delta$ be a $k$-derivation of $A$ into $N$: indeed, for $x, y$ in $A$ and $\lambda$ in $k$, one has the relations

the last equality resulting from the fact that $N$ is of square zero. The proposition follows from this.