Induct is what groups do when they formally accept new members, like the honor societies that, every school year, induct students who meet the qualifications.
When you induct someone into a group, you formally welcome him or her as a member. The word is typically used to describe official acceptance of a position or admittance into an organization, such as a professional society. It can also be used to describe someone who has formally enrolled in the military. Don't use it for informal joining, though — you wouldn't induct a new friend or be inducted at the gym where you have just started a membership.