1560s, from French inauguration "installation, consecration," and directly from Late Latin inaugurationem (nominative inauguratio) "consecration," presumably originally "installment under good omens;" noun of action from past participle stem of inaugurare "take omens from the flight of birds; consecrate or install when omens are favorable," from in- "on, in" (from PIE root *en "in") + augurare "to act as an augur, predict" (see augur (n.)).
INAUGURATIO was in general the ceremony by which the augurs obtained, or endeavoured to obtain, the sanction of the gods to something which had been decreed by man; in particular, however, it was the ceremony by which things or persons were consecrated to the gods .... If the signs observed by the inaugurating priest were thought favourable, the decree of men had the sanction of the gods, and the inauguratio was completed. [William Smith (ed.), "Dictionary of Greek and Roman Antiquities," 1842]