Nonsolvable groups with three nonlinear irreducible character codegrees

For an irreducible character $χ$ of a finite group $G$, the codegree of $χ$ is defined as $|G:\ker(χ)|/χ(1)$. In this paper, we determine finite nonsolvable groups with exactly three nonlinear irreducible character codegrees, and they are $\mathrm{L}_2(2^f)$ for $f\ge 2$, $\mathrm{PGL}_2(q)$ for odd $q\ge 5$ or $\mathrm{M}_{10}$.

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