What is the default transformation method to project data from from GCS_WGS_1984(WIKI 4326) to WGS_1984_Web_Mercator_Auxiliary_Sphere(WIKI 3857)?
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There is none because both coordinate reference systems (CRS) are based on WGS 1984. No geographic/datum transformation is needed.
If you happen to use the "old" Web Mercator version, "WGS_1984_Web_Mercator" aka EPSG::3785 or Esri::102113, the geographic CRS is sphere-based and you should use WGS_1984_Major_Auxiliary_Sphere_To_WGS_1984.