Presentation of a semidirect products of $N$ by $H$ can be written from presentations of $N$ and $H$. But for other extensions of $N$ by $H$ (cyclic, central etc.), which are not semi direct products, can we write presentation of the extension from presentation of $N$ and $H$?

To start with, following Alex's comment above, I think you need to look at some books on cohomology of groups to specify how the extension is going to be given. That means looking at the Schreier theory. Many years ago the Schreier theory of extensions was adapted to give exactly this by Turing.(Little known paper of his.) Ronnie Brown and I gave a more modern treatment of it in a paper in the Proceedings Royal Irish Academy.(On the Schreier theory of nonabelian extensions: generalisations and computations, (Proc.Royal Irish Acad., 96A, (1996) 213227.)) That is quite general, but a simple version of the question can found discussed in many books on combinatorial group theory, such as D. L. Johnson, 1980, Topics in the theory of group presentations , number 42 in London Math. Soc Lecture NotesMS Lecture Notes, Cambridge University Press. and D. L. Johnson, 1997, Presentations of groups , volume 15 of London Mathematical Society Student Texts , Cambridge University Press, Cambridge. The advantage of these is that they do not require an imense expenditure of time to get to the heart of the problem. (An interesting follow on is to examine ways in which to start with an extension of groups, plus resolutions of the two ends and give a resolution of the middle term.) 

