On the average size of 3-torsion in class groups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> -exten

The Cohen–Lenstra–Martinet heuristics lead one to conjecture that the average size of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -torsion in class groups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> -extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> for permutation groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> for a broad family of permutation groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> , including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>5</mml:mn> </mml:msub> </mml:mrow> </mml:math> . We extend their results to prove that the average size of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -torsion in class groups of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>≀</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> -extensions is finite for any nilpotent group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> .