You will need the Full Write Groups permission or be and Admin of the group to perform this action.
You have 2 options when removing an individual from a group:
1 - Deleting the person
2- Adding a drop date
If you want to keep a historical record that someone was at some point in time a member of a particular group then you will need to add a drop date.
Option 1 - Remove group member by deletion.
First navigate the group, then select the drop down arrows located next to the Edit button and select "Delete".
Note: this will delete any history that this person was ever connected to this group.
Option 2 - Remove group member by adding a drop date.
First navigate the group, then select the edit button.
On the pop-up screen add the date dropped in the appropriate field.
These two actions can also be performed on an individual's profile.
First, navigate to the person's profile, select the involvement tab, then the group. From there you can select the Remove button to delete the person from the group or Edit to add a drop date.