In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme".

Hence my question:

Given an algebraic space $X$, is it true that there exists a scheme $S$ and an action of a finite group $G$ on $S$ such that $X=S/G$ ?

If I remember correctly, every algebraic space is the quotient of an affine scheme by an étale equivalence relation, so I tend to think that there could exist such equivalence relations that are not "implemented" by a group action...