Finite groups of non-prime-power order with exactly four character codegrees

For an irreducible complex character 𝜒 of a finite group 𝐺, the codegree of 𝜒 is defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>G</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mi>ker</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo>/</m:mo> <m:mi>χ</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> G:(χ)/χ(1) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ker</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>χ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> (χ) is the kernel of 𝜒. In this paper, we give a detailed characterization of finite groups of non-prime-power order with exactly four (irreducible) character codegrees.