Automorphism groups of spaces of minimal rational curves on Fano manifolds of Picard number $1$

Let $X$ be a Fano manifold of Picard number $1$ and $M$ an irreducible component of the space of minimal rational curves on $X$. It is a natural problem to understand the extent to which the geometry of $X$ is captured by the geometry of $M$. In this vein we raise the question as to whether the canonical map $\operatorname {Aut}_o(X) \to \operatorname {Aut}_o(M)$ is an isomorphism. After providing a number of examples showing that this may fail in general, we show that the map is indeed an isomorphism under the additional assumption that the subvariety of $M$ consisting of members passing through a general point $x \in X$ is irreducible and of dimension $\geq 2$.