Under your good auspices , i.e. through your influence, or the influence of your good name. In Rome only the Commander-in-Chief was allowed to take the auspices of war. If a legate gained a victory, he was said to win it under the good auspices of his superior in command.
“Auspex” is from avispex (avis and spicio), one who observes the flight, etc., of birds.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894