I assume in this question that if $G$ is a group extension of $N$ by $K$ then $N$ is the normal subgroup in $G$.
Let $G$ be a group extension of $N$ by $K$ where $N$ is the direct limit of a system of groups $N_i$. Can this be considered as the direct limit of group extensions of the $N_i$ ?
What happens for example with $BS(1,2) \simeq \mathbb{Z}[\frac{1}{2}] \rtimes \mathbb{Z}$ ?