Reading some of my old notes, I came across a remark, I don't understand. Summarized:

Let $(M^n,g)$ be an open riemannian manifold (in that case it came with sectional curvature $K\ge0$, but I don't think this is necessary here). One way of ricciflowing this manifold is to consider the universal cover $(\tilde{M},g)$ and find/construct a ricci flow $(\tilde{M},g(t))$, which is invariant under the isometry group $Iso(\tilde{M},g)$. Then this ricci flow descends to $(M^n,g)$.

How does this "descent" work and why do you need the invariance under the isometry group?

Regarding the last part of this question: According to Brett L. Kotschwar $Iso(M,g(0))=Iso(M,g(t))$ holds true for any ricci flow $(M,g(t))$ of uniformly bounded curvature. Therefore a further question is: Is this invariance at least under the assumption of uniformly bounded curvature unnecessary?