Polynomial-time isomorphism test for $k$-generated extensions of abelian groups

The group isomorphism problem asks whether two finite groups given by their Cayley tables are isomorphic or not. Although there are polynomial-time algorithms for some specific group classes, the best known algorithm for testing isomorphism of arbitrary groups of order $n$ has time complexity $n^{O(\log n)}$. We consider the group isomorphism problem for some extensions of abelian groups by $k$-generated groups for bounded $k$. In particular, we prove that one can test isomorphism of abelian-by-cyclic extensions in polynomial time, generalizing a 2009 result of Le Gall for coprime extensions. As another application, we give a polynomial-time isomorphism test for abelian-by-simple group extensions, generalizing a 2017 result of Grochow and Qiao for central extensions. The main novelty of the proof is a polynomial-time algorithm for computing the unit group of a finite ring, which might be of independent interest.