Let $X,Y$ be "spaces" on which a group $G$ acts. Typically I see the notation $X\times_G Y$ to denote the quotient of $X\times Y$ by the relation $(x,y) = (gx,gy)$.
In a paper I now see the notation $X\times^G Y$, without explanation, though it is in the same context - $G$ acts on $X$ and $Y$.
Are both notations standard/common (for the quotient $(x,y) = (gx,gy)$)?
EDIT: The paper is Hinich-Vaintrob Augmented Teichmuller spaces and orbifolds. Remark 3.1.3(i)