  Let $H$ be a nonabelian finite simple group. Huppert's conjecture asserts
that if $G$ is a finite group with the same set of complex character degrees as
$H$, then $G\cong H\times A$ for some abelian group $A$. Over the past two
decades, several specific cases of this conjecture have been addressed.
Recently, attention has shifted to the analogous conjecture for character
codegrees: if $G$ has the same set of character codegrees as $H$, then $G\cong
H$. Unfortunately, both problems have primarily been examined on a case-by-case
basis. In this paper and the companion [HM22], we present a more unified
approach to the codegree conjecture and confirm it for several families of
simple groups.
