Normal covers of 2-arc-transitive graphs of prime-power order

In a paper by Cai Heng Li in Bull. London Math. Soc. 33 (2001), it was suggested that ‘non-basic’ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -arc-transitive graphs of prime-power order that occur as normal covers of smaller <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -arc-transitive graphs might be rare and difficult to construct. This note describes some of the background to Li’s suggestion, and gives some examples of small valency, and then goes on to show that in fact there are infinitely many examples of valency <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> , for every integer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . It is also noted that the hypercubes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> even, together with a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -arc-transitive group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> of index <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">Aut</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>Q</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , show that the claims of Corollary 1.2 in the above paper by Li are not quite correct.