The mass-market beauty company had $96.8 million in sales for the first half of 2016.
Rights groups applaud the decision as a victory for individual and religious freedom.
He reports to chief executive officer Michael Buckley.
Verma was most recently chief information officer at Vineyard Vines.
Cooper reports directly to ceo Yehuda Shmidman.
Engelhaupt had worked at Under Armour since 2014.
Mike Pardini succeeds Pisani as president of the licensed sports group.