Let $M=G/K$ be a $G$-homogeneous manifold and suppose that $E\to G/K$ is a homogeneous (complex) vector bundle, i.e. it is defined by a representation $\phi : K \to \text{Aut(V)}$ for some complex vector space $V$. More precisly, $E=(G \times V)/ \sim$, where $(gk,v) \sim (g,\phi (k) v)$.

As usual, the projectivisation $\mathbb P (E)$ of $E$ is defined to be $\mathbb P (E) := (E\setminus 0)/ \mathbb C ^*$. Clearly, the group $G$ acts on the manifold $\mathbb P (E)$ and if we endow $\mathbb P(E)$ with the metric induced by $E$ and $M$, this action is by isometries.

Question 1): When is $\mathbb P (E)$ a $G$-homogeneous manifold, i.e. when is the group action of $G$ transitive? It seems sufficient to assume that the action of $K$ on $V$ given by $\phi$ is transitive, but is this actually equivalent to the $G$-homogeneity of $\mathbb P (E)?$

More concretly, I am interested in the case when $M=G/K$ is a compact Kähler homogeneous space and $E=T^* M$ is the cotangent bundle of $M$.

Question 2): Is there a criterion when $\mathbb P (T^*M)$ is $G$-homogeneous? For example, is this true if $M$ is a compact Hermitian symmetric space?

Could it be possible in this case, that the irreducibility of the corresponding representation $\phi$ is not only necessary, but also a sufficient condition?

I would guess that the answer to these questions is well-known, but, unfortunately, I could not find any reference, where this is proven. So I would be greatful for any help.