I am looking for methods to compute the conjugacy classes of any finite extension group N.G from the classes of G.

1$\begingroup$ It's hard to imagine any useful answer in this generality. For example, the answer will be different for the same $G$ and $N$, but different extensions $N.G$. $\endgroup$ – Frieder Ladisch Nov 14 '16 at 14:53
There are wellestablished 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 pgroups. 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), SpringerVerlag.
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.


$\begingroup$ You can do that by applying the method for elementary abelian groups twice. $\endgroup$ – Derek Holt Nov 14 '16 at 20:10
