Symmetry (killing vector) preserving diffeomorphisms

[center o bass]

Suppose that on a Riemannian manifold (M,g) there is a killing vector such that
$\mathcal{L}_{\xi} g = 0.$

How would one then characterize the group of diffeomorphisms $f: M \to M$ such that

$$\mathcal{L}_{f^* \xi} (f^*g) = 0?$$

How would one describe them? Do they have a name and can an explicit form be found?

 

[center o bass]

Alternatively, given a killing vector $\xi$ how would one describe the diffeomorphisms $f: M \to M$ such that $\xi$ remains a killing vector also for $f^* g$? I.e. $\mathcal{L}_{\xi} f^*g = 0$.