Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups

Given an almost simple group $A$, we algorithmically show that the character table of $A$ determines whether or not the Sylow 3-subgroups of $A$ are 2-generated. We show this property is equivalent to a condition involving the Galois action on characters in the principal $3$-block. This would be a consequence of the Alperin-McKay-Navarro conjecture.