Were you supposed to provide only one word for each category. Because I can think of many, and they're all likely to come from the same Latin root.
Noun: act, action, activity, acting, activism, actor
Verb: act, activate
Adj.: active, activated
Adv: actively (and here I'm stumped)
Actively is derived from active, so the noun for actively is probably action.
As to the verb, can't it be associated with both verbs?