Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product $P:=X\times_SY$ be given the structure of a $G$-scheme in a natural way, i.e. such that the projections $P\to X$ and $P\to Y$ are equivariant? I have searched through Mumford/Fogarty/Kirwan for a while now and could not find any such statement.

Is it false in general? Are there certain conditions under which it becomes true? I am usually only dealing with varieties over an algebraically closed field and my group is about as close to $\mathrm{Gl}_n$ as you like, but I need the fiber product over a third variety in this case.

Let $G$ be an algebraic group. Let $X$ and $Y$ be $S$-schemes such that $X$, $Y$ and $S$ are $G$-schemes and the structural morphisms are equivariant. My question is: Can the fiber product $P:=X\times_SY$ be given the structure of a $G$-scheme in a natural way, i.e. such that the projections $P\to X$ and $P\to Y$ are equivariant? I have searched through Mumford/Fogarty/Kirwan for a while now and could not find any such statement.

Is it false in general? Are there certain conditions under which it becomes true? I am usually only dealing with varieties over an algebraically closed field and my group is just the multiplicative group of the ground field, but I need the fiber product over a third variety in this case.