Temporize means "to compromise," "to act evasively so as to gain time or advantage," "to postpone action or a decision": "We need relief; don't temporize any longer." "Many a public official has found that he must temporize if he wishes to hold his office." Extemporize means to "speak with little preparation," "to say something without advance notice of having to speak": "A glib speaker, Senator Goff extemporized for nearly an hour." Temporize applies principally to actions, extemporize to words.
Dictionary of problem words and expressions. Harry Shaw. 1975.