Let $A$ and $D$ be two non-trivial abelian groups and $B,C$ be two non-abelian groups. Also, let $C$ is a free group and acts on $A,B,D$.

Let $0\to A \xrightarrow{f}B\xrightarrow{g}C\to 0$ be a short exact sequence of groups in which the action of $C$ commutes with maps $f$ and $g$ ($C$ acts on itself trivially). Also, let there exist homomorphisms $h_1 :D\to B$ and $h_2 :B\to D$ (commuting with the action of $C$) so that $h_2 \circ h_1 =1_D$.

If $A$ is a free $\mathbb{Z}C$-module, then is $D$ a projective $\mathbb{Z}C$-module?