The phrase "act genuinely" is correct and usable in written English.
You can use it when encouraging someone to be authentic or true to themselves in their actions or behavior.
Example: "In order to build trust with your team, it's important to act genuinely and show your true intentions."
Alternatives: "be authentic" or "act sincerely".
It is just as naïve to think that extremist Muslim groups act in the service of their religion as it is to believe that Christian extremist groups such as the KKK act genuinely in servitude of Jesus Christ.
We are called to hunger and thirst for justice, to be merciful, and to act genuinely rather than for show.
Act genuinely interested.
Act genuinely interested to keep them going.
Act genuinely excited, and don't be afraid to exaggerate things; use colorful descriptions and hand gestures.
If they confront you about it, act genuinely sorry so they won't have reason to hate you.
Fifteen years later, Mr. Benjamin acts genuinely thrilled by any suggestion that his candidacy is being taken seriously by people outside of his relatively small circle of supporters.
But the host, Steve Doocy, acted genuinely annoyed at being chastised, and told Mr. Wallace that if he still loved them, he had "an odd way of showing it".
Jake Gyllenhaal, who through a fluke of marketing and contest machinations has been something of a forgotten man this awards season – the Bagger thinks he should get some kind of recognition for his Santa hat dance in "Jarhead" – acted genuinely surprised when the supporting actor award came his way.
But beyond the visceral thrills on display, Bird and company know exactly when to go for a joke and when to stay serious, and they know that nothing makes an action sequences as convincing as when the actor looks and acts genuinely terrified to be doing it.
For me it means a willingness to deal with the root causes of things, to think and act in genuinely fresh ways.
