E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
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.