Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and $$\mathcal{L}_G:=\bigoplus_{k\geq 1}\mathcal{L}_G(k).$$ Then $\mathcal{L}_G$ has a graded Lie algebra structure induced from the commutator bracket on $G$.
Suppose that we have an exact sequence$$1\to A\to B\to C\to 1,$$ where $A,B,C$ are groups. Can we conclude that the associated sequence of Lie algebra $$1\to \mathcal{L}(A)\to \mathcal{L}(B)\to \mathcal{L}(C)\to 1$$ is also exact?
Furthermore, if we assume the exact sequence of groups is split, can we conclude that the associated sequence of Lie algebra is also split?