administer has 12 distinct synonyms spread across 3 of its 5 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
work in an administrative capacity; supervise or be in charge of
“administer a program”
perform (a church sacrament) ritually
“administer the last unction”
direct the taking of
“administer an exam”
Synonyms, antonyms and the broader and narrower relations for administer come from Princeton University’s WordNet. Full sources and licences.