The phrase "make pretensions" refers to making false claims or presenting oneself in a manner that is not truthful. Some synonyms for this phrase include "feigning," "posing," "pretending," and "putting on airs." Feigning refers to faking or pretending to be something that one is not. Posing refers to presenting oneself in a way that is not genuine. Pretending refers to behaving as if something is true when it is not. Putting on airs means to act in a way that is pretentious or overly proud. All of these synonyms are useful for describing behavior in which someone presents themselves in a way that is not entirely accurate or truthful.