Coprime action and principal blocks

Abstract In the situation where a finite group A acts coprimely by automorphisms on another finite group G , we characterize when the principal p -block of G contains a unique A -invariant irreducible character, in terms of the subgroup $$\textbf{C}_{G}(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of fixed points under such action.