There are well-established methods for doing this when $N$ is an elementary abelian $p$-group for some prime $p$, which involve setting up affine actions of $G$ on $N$, regarded as a vector space over the field of order $p$, and computing its orbits. Since these actions are on the elements of $N$, the applicability is limited by $|N|$ but it can be done for $|N|$ up to about $10^8$ or perhaps $10^9$.

They were first described by Felsch and Neubüser for finite $p$-groups in:

V. Felsch and J. Neubüser. An algorithm for the computation of conjugacy
classes and centralizers in p-groups. In Edward W. Ng, editor, Symbolic and
Algebraic Computation, volume 72 of Lecture Notes in Comput. Sci., pages 452–
465, Berlin, Heidelberg, New York, 1979. (Marseille, 1979), Springer-Verlag.

But they really only depend on knowing the conjugacy classes of $G$, and they are described for general finite groups in Section 8.8 of

D.F. Holt, B. Eick and E.A. O’Brien. Handbook of Computational Group Theory
Chapman & Hall/CRC, 2005.

Note that, for finite groups with structure $M.G$ where $M$ is a solvable normal subgroup, you can get the classes of $M.G$ from those of $G$ by repeatedly applying the method for an elementary abelian normal subgroup. The currently best known algorithms for finding the conjugacy classes of a general finite group start by doing for the quotient modulo the largest solvable normal subgroup.