Let $G$ be an algebraic group over an algebraically closed field $F$ of characteristic 0. Let $r>0$ be an integer. I want to express the size of the automorphism group $Aut(G^r)$, where the automorphisms are in the category of algebraic groups over $F$, in terms of $Aut(G)$, which we assume to be finite. Here $G^r = G \times ... \times G$.
Is the formula $\# Aut(G^r) = r! \cdot \# Aut(G)$ correct?
I think that the only way to get an automorphism of $G^r$ is to combine swapping factors ($r!$ ways to do so) with some automorphism of $G$, but I'm not completely sure about this.