When does a representation of a compact Lie group extend to an extension of the group?

Question

Let $G$ be a compact simple Lie group, let $H$ be its (finite) group of outer automorphisms. I'm interested in the semidirect product $G' = G \rtimes H$.

Let $\rho : G \rightarrow U(n)$ be an $n$-dimensional unitary representation of $G$ (not necessarily irreducible).

Let $A_{\rho}$ be the set of representations $\rho ' : G' \rightarrow U(n)$ of $G'$ that extend $\rho$, i.e. such that $\rho' |_G = \rho$.

I would like to know if the set $A_{\rho}$ has been studied, any reference is welcome. More precisely, I would like to know

1. When do we have $A_{\rho} \neq \emptyset$ ?
2. Is $A_{\rho}$ finite ?
3. If $A_{\rho}$ is finite, what is its cardinal, as a function of $\rho$ ?

If things are simpler in a particular example, one can take $G=SU(N)$ with $N \geq 3$ and $H = \mathbb{Z}_2$.