Extending huppert’s conjecture from non-abelian simple groups to quasi-simple groups

We propose to extend a conjecture of Bertram Huppert [Illinois J. Math. 44 (2000) 828–842] from finite non-Abelian simple groups to finite quasi-simple groups. Specifically, we conjecture that if a finite group G and a finite quasi-simple group H with Mult(H/Z(H)) cyclic have the same set of irreducible character degrees (not counting multiplicity), then G is isomorphic to a central product of H and an Abelian group. We present a pattern to approach this extended conjecture and, as a demonstration, we confirm it for the special linear groups in dimensions 2 and 3.