(verb) reprove; exhort
This word has contained the notion of "warning" since it arrived in English (from Latin via French) in the 14th century; but its chief modern meaning, to "warn against error or fault", is an 18th-century development.
It is important to forget this sense of reproof when listening to Pucelle asking for supernatural help: "Now help I ye choice spirits that admonish me" (Henry VI Part 1, V.iii.3). Why should she be asking spirits for help if they routinely tell her off? Of course the word does not have this nuance: it just means "forewarn, inform". And it means simply "warn" when Henry talks of the French acting as their consciences "admonishing That we should dress us fairly for our end" (Henry V, IV.i.9). The same point applies to the noun admonishment. In Henry VI Part I (II.v.98), Mortimer tells Richard to be careful, and he replies: "Thy grave admonishments prevail with me". Similarly, Andromache asks Hector: "When was my lord so much ungently tempered, To stop his ears against admonishment?" (Troilus and Cressida, V.iii.2). Here, too, the word means only "warning".