Let $G$ be a group and let $H$ be a factor group of $G$. Is there any result that relates $\operatorname{Aut}(G)$ (the automorphism group of $G$) and $\operatorname{Aut}(H)$?

As a very special case of the question, let $F_2$ be the free group with two generators $x$ and $y$. Let $G_2$ be the factor group of $F_2$ by adding relations such that $[x,[x,y]]=[y,[x,y]]=1$; that is, $G_2$ is the discrete Heisenberg group. Then is there any relation between $\operatorname{Aut}(F_2)$ and $\operatorname{Aut}(G_2)$? Hence or otherwise, how to find out $\operatorname{Aut}(G_2)$?