Extensions of groups with solvable word problem

Question

Is there a reasonably simple (to describe) example of a short exact sequence of finitely generated groups $1\rightarrow N\rightarrow G\rightarrow H\rightarrow 1$, where $N$ and $H$ have solvable word problem, but $G$ does not? I assume that solvability of the word problem is not closed under extensions, although for some reason I cannot find anywhere an explicit reference. So I would be happy to see some counter-example.