Let $H$ be the kernel of an action of a group $G$ on a space $X$. Is there a term for the actions with the property that the action of the quotient group $G/H$ on $X$ is free? 



Actions that become free after quotienting out their kernel
Yes, it's called **effectively free** (see for instance [here, page 3](http://www-users.math.umn.edu/~olver/di_/rel.pdf), Fels/Olver, *On relative invariants*).


